We went to Washington DC in 2014. We stayed at the Embassy Suites. Everything walking distance. Georgetown 10 minutes walk, White House 20minutes, zoo 30 minutes, but there are two metro stations 5 minutes away. Great hotel with pool, fitness room, breakfast and evening drinks (5.30-7.30pm) and snacks included in price.
We did a 2 centre holiday with 6 days in Washington and 5 days in New York. We did NYC first, flying into JFK and then using mega bus to travel between NYC and Washington. This was the bargain of our holiday...$3 each for the 4 hr journey by luxury coach. Glad we did Washington 2nd, it was so relaxing after NYC.
Do get in touch with British embassy in Washington as they can get you tickets to tour White House, they advised us how to tour pentagon and Capitol too. The Smithsonian museums and Washington zoo are brilliant and free. We found the holocaust museum excellent too. We paid a dollar to reserve tickets for the Washington monument online, this saved queuing at 8.30 am for free tickets...well worth the lie-in if you ask me!
Our best find was a company called 'free tours by foot'. They do tours of Washington: the Mall, Arlington cemetery etc. for free, you just pay a tip at the end. The guides are local and brilliant. We took 3 tours with them after deciding not to do the drive past open top bus tour. Google them they have fab reviews.
Enjoy your trip!