Please view the map from priceline first.
http://www.priceline.com/hotels/Lang/en-us/images/regionmaps/65.gif
You'd better pick up the hotel in two areas, Crystal City(Reagan National Airport) and downtown. Each big airport in US has public transportant to all the part of city. And DC's airport is just 5 miles from all of the monuments. At least you can take their hotel bus.
And also they have 2-star hotel just for $35/night in Crystal City. Please check the link. http://www.priceline.com/hotel_savings/WashingtonDCHotel.asp
The other choice is the hotel in downtown area. You would even walk to white house, but they must be very expensive.
Good luck.