I think you might get most out of an itinerary that starts in Barcelona and ends in Paris. Don't worry about adding another location after Paris -- give yourselves time to enjoy those two cities. Not sure why you settled on Nice in between -- you might find more variation if you picked a place like Avignon or another Provencial town. As you are from Florida, I don't imagine your are that curious about the seaside in Nice -- the beaches are not nearly as lovely as in your home state. With a solid two weeks, you could have 4 nights in Barcelona, 3 or 4 nights in Provence (rent a car to get the most of your trip), and another 4 nights in Paris. You will lose one night flying over there, and another full day flying home....plus your first day will be limited a bit due to jet-lag, but at your ages, and in a place like Barcelona, you will be able to push through it!

Seems good to me - 4 days Barcelona and 3 in Nice are about right for most folks - you can take an overnight train from Nice to Paris to say a half day time and save on the cost of a hotel to boot.
Only 2 days in London is frustrating for that huge city with so so many famous sites - Amsterdam is much smaller and has fewer blockbuster sites and is just a nice city to relax it - a favorite with younger folk from all over the world - especially those who patronize its world-famous coffeeshops where cannabis is sold over the counter legally.
London is just two hours by Chunnel train however and may offer better flights back to Miami. Personally I'ad ax a day off Barcelona and add it onto London or Amsterdam.
For lots on trains I always spotlight these IMO superb sources - www.seat61.com; www.rikcsteves.com and www.budgeteuropetravel.com. For discounted French TGV or night train tickets - www.voyages-sncf.com - early birds get the worm there as you can save o ton of money but these discounted ducats are sold in limited numbers and often sell out early. Also check www.idTGV.com for cheaper TGV tickets.
For Chunnel train tickets to London same story - www.eurostar.com and for Thalys high-speed tickets Paris to Amsterdam - same story - early bird can get way way cheaper tickets - www.thalys.com or www.voyages-sncf.com.
There is a France-Benelux Railpass but it is not a good deal for your plans - in part because to use it on Thalys trains you incur a steep surcharge that actually can cost more than some of the deep discounted tickets you can get on www.thalys.com.

I'll take the middle road and suggest BCN, Nice and Paris, but stop there. Don't discount Barcelona - it's a great place that some experienced Europhiles think is superior to Paris. We were in both cities last May and have been in fall as well; weather's always better in Barcelona!
EasyJet has flights from BCN to Nice every day except Tuesday and Vueling has two flights daily.