We were in DC in the summer of 2014 for 10 days. There is so much to do and we only scratched the surface. Here are a few things I learned that might be helpful to you.
We flew into Dulles, rented a car, and drove to Arlington. We bought some snacks and drinks, dropped off the luggage, food, drinks, and teenagers at the hotel before returning the car to Reagan. Alamo did not charge us to return the car at a different location. Our hotel sent a shuttle to pick us up from Reagan but we could have easily hopped on the Metro. We stayed at the Embassy Suites in Arlington because it was cheaper than staying DC and we could easily take the Metro. We have stayed in a dozen Embassy Suites around the country and Arlington is still our favorite one (The free breakfast area is well done and spacious; breakfast is plentiful; and there are shelves in the closet to keep our stuff organized). The hotel does provide shuttle back and forth to the Metro stop at set times. But, if you will be taking Uber/Lyft, it may be better to stay in DC.
On the day we went to Mt. Vernon, the hotel dropped us off at Reagan to rent a car. We enjoyed Mt. Vernon, but once again, it is a lot of walking on his estate. We returned the car at the end of the day to avoid paying for parking. And we rented again from Reagan to get back to Dulles. The rentals to and from Dulles gave the space geeks in our family time at the Air and Space Museum at Dulles. This is home to the Space Shuttle Discovery. There is a parking fee.
Pay a few bucks per person to reserve tickets online ahead of time. That way, you have control over your time in advance and are not stressed everyday to go get tickets. Plus, we didn’t want to go line up early in the morning for tickets. It didn’t have to be like RDing at WDW!
DO NOT drive in DC! Ditch the car. Hotels charge a lot for parking and it is not that convenient to have a car in DC (parking and traffic). Depend on Uber/Lyft. It’s probably the cheaper alternative.
There is a lot of walking; from and to Metro, between the monuments and museums, and in the museums. With mobility challenges, Uber/Lyft may be the way to go. Choose the museums people want to see. Some places, we only wanted to see a couple of items and then we left.
Go during a time when there are not crowds of people pushing and shoving. Summer was hot and crowded.
Do not plan 2 building tours in a day. It was stressful to get from one place to another on time and it didn’t leave much time to stop and smell the roses. We rushed through the National Gallery of Art -Sculpture Garden. It was a cool place to stroll, sit, and chillax. Places are far away and lots of walking!
We toured all of the monuments on bike by ourselves, took Metro and walked to Arlington and the Marine Corp Memorial, and did a guided evening bike tour. But with 2 elderly parents and a special needs son, you might want to consider a narrated bus tour with opportunities to stop and get off the bus. They have day tours and night tours of different lengths. In San Francisco, I’ve seen bike tour guides with a “buggy” for 2 in the back. I don’t remember seeing them in DC and couldn’t find any online. I did find this and may be reasonable and personal for your family: https://www.viator.com/tours/Washington-DC/Capitol-Hill-and-DC-Monuments-Tour-by-Electric-Cart/d657-5713UNVEILED
My family also enjoyed the 9/11 Pentagon Memorial. It was very solemn and beautiful. It was a long walk from Metro. Uber/Lyft
If the Washington Monument is open again when you go, reserve a tour to go to the top. You get a 360 view. We got tickets near sunset.
For the White House tour, you need to get your Congressperson to get you a reservation. Do that ASAP. It takes months and they don’t let you know your reservation time and date until like 4-6 weeks before your tour.
You can get your Congressperson to get you a Capitol tour, too. But you can book this online and select a time that best meets your need. Do this early, too.
We also enjoyed a free tour of the National Archives. But I recall going up and down stairs. They have lovely exhibits on our nations history, including some paintings of Washington. https://museum.archives.gov/reservations-and-tours
I took several sheets of paper, made a row for each day we were going to be in DC, and taped it to a wall. Each family member wrote must see/must do on Post Its, one Post It for each thing. We moved the Post Its around to fit the tour schedules.
We enjoyed eating at Legal Seafood, Matchbox, Founding Farmers, and Sweetgreens (seating here is not plentiful at some locations around DC). Founding Farmers requires reservation days in advance or show up at opening.