Stay in Arlington or Rosslyn. Trust me. The Sheraton National was nice, offered a shuttle that ran constantly, and was far more affordable than anything downtown. Last August I stayed at the Hyatt in Rosslyn which I didnt like as much (they charge for wi-fi access, and there's no 4g signal in the building. Little things like that they nickel and dime). But the rooms are nice and the metro is literally 1.5 blocks away.
As far as what to do....that's gonna depend on your interests. SOme people could go for 2 or 3 days and be pretty content with what they've seen. Some people would be interested in a lot more detail. A week is probably just fine.
Definitely visit the Air and Space museum. An absolute must for nerds. If you can manage it, also go see the satellite facilty at the airport. Seeing the Enola Gay and the Space shuttle was just staggering.
Also, hang out in Georgetown. Check out the argentinian gelato place. Best I've ever had. Walk around and just chill there. Great vibe in Georgetown. Walk down to M street if you're a horror movie fan.
I've never taken the tour of the White House, but I have in the Capitol WAAYYY back (before 9/11) and it was kinda cool but nothing amazing. Being on the lawn is impressive enough.
Be prepared to wait in line for some of the coolest stuff. Like the Declaration of Independence.