many years ago I toured CA with my cousin. We did not have a car and used public transportation for the entire time. While we managed pretty well in San Fransisco and San Diego, I would never do LA again without a car. Public transportation in the LA area was a nightmare. We spent more time on buses than touring. With your limited time frame, I would either skip LA, rent a car, or stay one night in LA in a location convenient to what you want to do there.

speckles-I really don't know what to say after reading your reply. You asked for advice and we gave it to you. But the reality is that unless you stand up to your friend (which I would not call a dear friend)this trip to the USA is going to be miserable for you.

You said that this trip was your idea yet your friend is planning the whole thing. I am sure you will enjoy Washington DC but you are spending more time there because she wants to. You are staying in Anaheim because she wants to. And think about it, she doesn't like LA because she saw it when she was 15! If someone said that to me I would just laugh. My daughter hated London when she was 16 because she was a teenager and wanted to be home with her boyfriend. Now she is 20 and says she regrets her attitude.

You need to get some courage and stand up to your friend. Since she is getting her way for Washington DC, you should get your way in LA. Stay 2 nights in Anaheim and then move to Hollywood. From there you can jump on the Metro and in one stop be at Universal Studios (no long bus ride). And although I would recommend a car, you can definitely get around LA from that location on public transportation.
I don't understand why you can't change the hotel. You aren't coming until August. Cancel 3 of the nights and switch hotels. If your friend can't compromise that much, you are in for a long and miserable vacation.

Please have a talk with your travel companion and let us know what happens.

I'd suggest moving to downtown LA for part of the trip. Especially if you go on a weekend, you can find a good rate on the discount sites such as Hotwire. EG, a recent poster just got the Omni for just over $100/nt. There is definitely enough to keep you busy downtown for a day, eg Little Tokyo, Olvera Street, Broadway, City Hall, Disney Concert Hall, Chinatown galleries, museums. Thursday night is a good night to spend Downtown, especially around Little Tokyo. Also, you will be at the hub for the light rail and rapid bus system. The Red Line light rail can get you to Hollywood in 15 minutes. There's a flyaway bus to LAX from Union Station.

One day at each park in Disneyland is fine for adults. There is lots of live entertainment, eg Aladdin musical in Cal Adventure and Billy Hill in Disneyland. August will be very crowded, so go as soon as the parks open.

Yes downtown LA is another good suggestion if you use Priceline or Hotwire. With Priceline you can't guarantee two beds but if your friend refuses to switch hotels then you could probably afford the room by yourself. The Omni for around $100 is a great deal!!

Stay in Anaheim with your friend for the first 2 days.

For the 3rd day, you can take the special coach that goes from Anaheim to Universal, included in the price of an all-day Universal ticket. After a few hours at the park, you can walk 7 minutes to the subway for a 6 minute ride down to Hollywood. You can see the Walk of Fame, kodak Theatre, etc for 2 or 3 hours then hop back up to Universal for the last couple of hours and for your coach back to Anaheim. I think it's a 10am to 8pm deal, info is on the Universal website.
For the 4th day, book yourself a tour from Anaheim. Starline and Grayline and a few other companies have all-day tours through L.A. and Beverly Hills etc.
For the 5th day, you and your friend should move to a hotel in Santa Monica or Westwood for shopping and relaxation. (You could splurge on a hotel since you've been staying at HoJos for 4 nights!)
Have your hotel hold your bags on that final morning and take a bus up to see the fabulous Getty Center for a few hours. Pick up your bags around 2:30 -3pm and you'll have plenty of time to make it to LAX for your flight to Vegas since you're already on the west side.
If you feel adventurous and brave, you might even want to rent a car for the final day from your hotel area (with LAX dropoff). Buzz around Beverly Hills or take a drive out to Malibu, then down the coast to the airport.