I've gotten many behind the scenes tours of the Senate Bldg and Capital, with a White House tour, but I'm not sure how much the general public is privy too, without an inside connection. You might contact your Congressmen or Senators' offices to find out how they may be able to arrange some things. If nothing else, go find your Congressman's office and say hi. Get your kids pics taken with your Representative. Things like that leave impressions on kids.
Not on the Mall is the Spy Museum. Get a schedule for times -- long lines. I forget which street it is on, but it is a few blocks north of the Mall. Really cool. Arlington Cemetery is pretty cool, but it is a little out of the way. There is plenty to see on the Mall. Jefferson and Lincoln memorials are really impressive. Make sure you read the quotes. Holocaust Memorial, Art and History museums, Air and Space, botanical gardens are all good. Maybe see a play at Ford's Theatre. Skip the American Indian museum. It is so unorganized and the flow is terrible. Check out the Supreme Court. Oh yeah, I love the sculpture garden on the mall. The train station is pretty cool. The WWII Memorial is fairly new, and good for a sit by the fountains.
No guidance on where to stay. I always stay with fam. Leave the pretty shoes at home, and go for the comfy running shoes. Grab a back pack, and keep refreshed with gatorade. Plenty of drink vendors on the mall.
