I'd look into sublets for a room in a house. It'd probably be among the easier options for finding a short-term rental and would fit into your budget. Like terrence81 suggested, Dupont Circle, U Street, and Capitol Hill are all nice areas. U Street is a little grittier, Capitol Hill is a little quieter. Logan Circle is also a nice area that you could look into. If you really want to walk, Logan Circle would probably be your best bet, though it's still a little far.
13th & F is basically a commercial area, so there's not a lot of places to live there (and the few that are available are expensive and likely not short-term). Metro Center is nearby, so you'd want to aim for the red, orange, or blue Metro lines to get to work.
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