Mrs F and I returned from our American Adventure just last week. We used the Virgin LHR - IAD to get there and back but to avoid having to travel all the way out to IAD through the traffic, we did the domestic flights from DCA which is much closer and can be reached via the metro or, as we did, Uber in about 15 minutes from Capitol Hill. While we're on the subject of Uber, we found this to be absolutely invaluable. Mrs F broke her leg not so long ago and having a reliable way to summon a cab while you sit in a coffee shop rather than standing on the sidewalk was amazing, that and not having to have the right cash on you to pay - everything is charged to your card - was invaluable.
We used AirBnB for accommodation as we thought the Washington hotels were overpriced for what you get - watch out for crazy breakfast prices in hotels!
You've had lots of great advice already but I'll stress that pre-booking the Capitol Hill tour and Washington Monument tickets is essential. All you pay is around $6 postage and it saves you queuing for tickets that might not even be available by the time you reach the front of the line - Washington Monument tickets are usually all gone by 0830.
When you enter the Capitol Building for the tour you'll go through security and 2 sets of glass doors. Before you go downstairs into the huge lobby to start the tour, visit the desk just at the top of the stairs and ask for tickets to go into the House of Representatives and the Senate (you'll need ID to get these so bring a passport). Tourists are told about this at the end of their tour but if you grab them before you start you'll save yourself a walk and have your tickets before the are all gone.
If you visit any of the Smithsonian museums catering is pretty variable, Air and Space has a McDonalds if that's your thing and we thought that the Stars and Stripes in the American History museum was overpriced and low quality. A far better option is to walk over to the Museum of the American Indian, they are all pretty close so you can do this with ease, and eat at Mitsitam which is an award winning restaurant.
http://www.mitsitamcafe.com/content/menus.aspYou might want to think about getting a mobile phone pay-as-you-go sim card from 3. For £20 they do unlimited data and "feel at home" which lets you use your phone all over the States without worrying about the cost, it's something like 3000 included minutes which should be plenty. We were there for 3 weeks and this was possibly the best £20 we spent.
On our way home we spent the night at IAD since our domestic flights were on a different PNR to the TATL we didn't want to take the risk of missing it. This gave us the perfect opportunity to visit the Udvar-Hazy annex to the Air and Space museum. It's well worth a visit and every 30 minutes one of the docents (basically retired astronauts and air force pilots) starts a tour. Nominally these are 90 minutes long but these guys really LOVE their subject so you can easily get them off onto tangents talking about their trips to space.
Enjoy your adventure!