We stayed with my brother an hour away in Frederick, MD, so I don't know about where to stay.

As for things to see:
The White House if you can get in
Udvar-Hazy Air & Space museum at Dulles Airport (this one is 10 times better than the one on the Mall with about 200 planes/spacecraft including a space shuttle, Concorde jet, and the Enola Gay that dropped the bomb)
Arlington Cemetery (don't waste money on the bus ticket-just walk to the JFK gravesite and the Tomb of the Unknown soldier)
Bureau of Engraving (where they print money)
Natural History Museum
American History Museum
National Archives
The Capitol
Washington Monument (but I think its closed for a few years to fix the elevator)
Mount Vernon
DC Temple

We also enjoyed Fort McHenry in Baltimore (where the Star Spangled Banner was written by Key)