I think a lot of time, the British Isles are recommended as you don't have the difficulty of language. And my first thought was also Ireland (which happened to be my first Europe destination too).
But - truth be told, I don't think that Ireland really prepared us for subsequent trips to the mainland. Every country is new in its own way. And, Ireland/England presented unique challenges for Americans if they plan to self-drive at all.

I like the suggestion to sit down and think about why you've decided to go to Europe and use that as your guide. Language, driving, etc are all going to be things you have to handle anyway but going somewhere else first won't change that. You can learn some of the local language just like you can learn to drive on the left. And you'll find plenty of people speak some English throughout Europe. (our 2nd country in Europe included self-driving around remote Romania.... even that was fine.)

Five people and luggage for five people means a large car. I'd avoid a driving holiday. Trains are great in Europe and maybe a different way to travel for your family. Either way, pack light, research trains and you can all sit back and enjoy the ride. Public transport means avoiding very narrow roads or small villages you should never have entered and are now stuck. Loads of great train info on Seat61 website and Rome2rio.com will help with getting from A to B.

Not wanting to contradict myself, I just want to share my own experience but not give any advice so far.
one of my favorite places around Europe is south Germany, more precisely around Lake of Constance.
There is a lot to see and do, that will give a good insight of one of the many European cultures.
For the mountains, it is fun to ride gondolas and chairlifts. Also fun to cross several borders in one day combining bus, train and ferry or take a cruise on the lake. Parts of area are rural, many farms making cheese and wine yards in the German part. You can rent e-bikes on mountainous tracks.
Last but not least, several interesting towns or places are not too far, Munich, Zurich, Innsbruck, the famous castles (Linderhof, Neuschwanstein), baroque churches.
We stayed in the area several times. Unfortunately, our last stay was a disaster with rain almost every day...

5 people in early August is an issue especially when Edinburgh disappears and Paris goes due to parties.

If it were me, I'd look at one of
Scotland (excluding Edinburgh)
Brittany, fine a Gite and just get to know a small town, beach etc
Mosel, hire an appartment and get to know the river
All these places are interesting, lots of exercise, not cost prohibitive and not seriously hot, packed out with tourists etc.

How "abroad"do you want your experience to be?
The Uk is easy of course, apart from the "wrong side of the road" bit, and some of the accents, ditto Ireland. The Netherlands most people speak English to some degree, but it is still foreign enough to feel abroad. You do not have to stay in an expensive city like Amsterdam. It is a small country with good infrastructure. Great for cycling of course, and you can easily rent bikes. But also trains and roads are good.
France if you or kids speak any of the language, but avoid Paris. More French speak English than used to be the case.
Spain Portugal and Italy will almost certainly be scorchingly hot, and are suffering from droughts which will only get worse.
Germany is a big country and you could choose a single area to explore, the Moselle, the Rhine, maybe the north coast, or the German Alps.
The further east in Europe the more "abroad" it will feel, but costs will generally be lower.

As well as looking at gites/cottages/airbnb consider holiday parks and campsites with accommodation which may be cheaper, though nowhere will be cheap in August I am afraid, peak school holiday time for Europe.
Maybe you could look for a house swap or house sitting.

What about Iceland? Long days that time of year and if the kids are studying geography they would learn so much by osmosis

I would suggest that you consider the Netherlands, for the reasons mentioned by Hetismej. Most people speak English, it's a lovely country, and it is unlikely to be excessively hot. I would recommend staying in one of the smaller cities. Leiden, Delft, Utrecht, and the Hague are some of my favourites. The rail system is excellent, and you could easily visit Amsterdam from any of these cities. If you would prefer a larger city, I can heartily recommend Rotterdam, a lively and vibrant city that hasn't been discovered yet by mass tourism. It should ould be especially attractive for teens and young adults.

$10,000 sounds like a lot, but it's only $2000 per person. I would check airfares and then decide on a destination where your budget will be sufficient. England and Scotland are expensive, and Ireland isn't far behind.

Some people have said that southern Europe is not unbearably hot, and that at least it's not humid. I wonder how recently they've been in southern Europe. I live in Italy, and not even in the hottest region, but lately it's become unpleasant to spend many hours outdoors in the middle of the day. In recent years the humidity has also been much higher than we are used to. Southern Spain is almost always hotter than central Italy, where we live.

I can't understand why the language should be a criteria. I am barely able to align few words of spanish or german but I never found difficult to visit german or spanish speaking countries. It is part of the thrill to travel abroad, the sense of achievement when you manage or the big laugh when you picked something that looked delicious and tasted awfully bad.
it is lnot like visiting a remote part of Korea or Japan when you need to use an ATM and you have no clue how to do it (and I am referring to an era when smartphones were not there to help you to decipher).

There are many other things that are different when you are abroad, shopping or restaurant hours, do and don't, weather, food, social interactions... some of them more important than a common language.
and even small things may ruin the experience (a Chinese will get mad if he can't make a cup of tea at any time. Being a bit like that, I always travel with an electric kettle and some tea bags, just in case).

Restaurant hours and unusual foods are things you can (and should) read up on before your trip. However, a language is something that you can't master in preparation for a trip. You should learn a few polite phrases, but if you ask a question, you're unlikely to understand the answer.

Some people do well when they encounter people with whom they don't share a language, and others find it stressful. If they aren't accustomed to travel in Europe, visiting a country where a reasonable number of people speak their language reduces greatly the number of difficulties they're likely to have.

I've travelled quite a lot in countries where I don't speak the language, including to China, where I couldn't even make a reasonable guess at what the signs were saying. (This was over 20 years ago.) I've almost always managed to communicate, but there have been failures.

My first trip to a non-English-speaking country was a camping trip in northern Quebec. I needed fuel for my camp stove, so I went to a sort of general store, equipped with a little French/English dictionary and the memory of my student French course. I greeted the saleswoman, and understood her response. I asked politely for white gas. She hesitated, and then smiled and went away. She returned with a bottle of silver polish.

Once in China, I was lunching with a colleague. We tried to communicate that we wanted duck for lunch. When the waitress couldn't understand, I quacked. She understood and we all laughed. Then my colleague reminded me that we had seen lovely broccoli in the market, so I decided to push my luck. I drew a picture of broccoli on my notepad. We got duck and mushrooms for lunch.

These days, you can almost carry on a conversation using Google translate. The last time we were in Paris, the proprietor of our apartment said something, activated Translate, and we heard it in Italian. Then we replied to his phone in Italian, and he heard it in French. However, this isn't really the way to carry on a fluent conversation.

The Netherlands is ideal in that you can experience a foreign language, all around you, but nearly everyone is nearly fluent in English. I lived and worked there in the late 1980s, and even then nearly everyone spoke English. I got fairly comfortable with Dutch, but if I asked a question, often the other person responded in English.

I know that everyone wants the input from everyone going abroad but in my family there has never been much input in as to where we go. We have done 40-50 trips abroad over the last 25 years. My youngest daughter now has taken over planning much of the day to day details but otherwise everyone depends on me to make the decisions as to where we are going. It is more difficult to plan things now as everyone has their own careers but otherwise most of the heavy lifting is up to me. Maybe it's because I pay? If the OP can get input from others it would help narrow down their options. It does require however those with input to do some research.
I do agree that language should not be a barrier. Another language is part of the cultural differences. It is however helpful to learn some simple phrases like thank you. Many Europeans do speak some English.
I have not done an August holiday since I was in University but I would bear in mind that Europe among other places has had some 40 degree heat waves in the summer that would make travel uncomfortable. This includes places like England. Would try to book places with AC if possible. It is likely to be hotter in the south of Europe and the humidity is not low and likely to be around 60% and if there is a heat wave it may involve much of Europe. Lastly $2000 a person is not a lot of money. Once you find the airfares to places you are interested in you can see what you have left for day to day expenses. London is not a bad place to start but it is very expensive. So is Paris and obviously you should avoid Paris while the Olympics are on but it might be different at the end of August where many Parisians have left the city. Otherwise you have many options but finances may be major factor.

London/Amsterdam makes a great trip. Paris with the Olympics will not work this year and Edinburgh is very busy with the Fringe and other festivals. London is always great to visit but you will want to find a place with air conditioning in both places as not universal and summers are getting hotter. If you decide on Amsterdam, book tickets for the Anne Frank Haus as soon as your plans are firmed up. So much to see and do in both places. I was in London and Edinburgh this year in early October while my daughter and her boyfriend did a London trip and as she was able to get tickets to the Vermeer retrospective at the Rijksmuseum they had a busy two days in Amsterdam and took the train there from London. She had been to both places previously but her partner had not. They couldn't get tickets to Anne Frank Haus but they saw and did a lot in a short time.

I would say Paris should be top of the list but it's out due to Olympics. I am not a big fan of England and Scotland as IMO they are different from rest of Europe except London.

First, you need to determine what you want this trip to be. For example:

1) Major cities. If so I would say something like Rome, Amsterdam, Vienna is a good mix of things. Probably priciest of all the options I am giving. Maybe throw in Venice as the most unique city in Europe instead of one of the other 3.

2) Budget major cities: Budapest, Prague and Vienna would be my recommendation. English is widely spoken everywhere, they are close to each other so cheap and convenient travel with train or car. They are all large and beautiful European capitals with a lot of history and things to do and see. Finally, they represent the best value IMO.

3) Beach: as everyone mentioned August is hottest month in europe. So maybe hit the beaches in Spain and Portugal. They are also relatively cheaper compared to northern Europe. Personally August travel is fine if you don't mind the heat. I also don't get why people go to Europe for beach (from North America) when Carribean is better and cheaper for that.

4) One country trip: some people just like to stick to 1 country as it makes it easier to move around. I would say France but Olympics is an issue. Other good options are Spain or Italy in the south and Belgium or Netherlands in the north.

One thing to keep in mind is southern European countries are cheaper than those in the north. Eastern European countries are cheapest hence Prague and Budapest.

Look up videos about luggage. You don't want to be lugging around suitcases if you plan several places. My wife and I travel z
europe with carry-ons as long as you are willing to do laundry but it's worth it.

The OP has not returned so we are all probably just whistling in the wind (hasn't returned to other threads she's started either - so either she isn't really interested - or- she can't figure out how to find her way back?? )