This might be a bit difficult as most hotels are set away from the Mall; some are a couple of blocks at the closest but most of them are farther. You'll most likely end up taking the Metro to one of the stations closest to the Mall and if that's the case, you may want to base your hotel decisions on proximity to a Metro station rather than to the Mall itself.

Chinatown/Gallery Place is a nice area for tourists. So is Dupont Circle. Both have ample restaurants and shops to keep you busy in the evening. The Kimpton Hotel group has a nice bunch of properties in both areas. They will hit the upper limit of your price range though.