Do not know cost, but one of the closest hotels to the Smithsonian museums is the Holiday Inn Capitol Hill. It is almost right outside the back (non-Mall side) entrance to The Air and Space Museum and is centrally located as far as the length of the Mall. Also in a great location, but I am sure much pricier, are the JW Marriott and The Willard hotels. BTW, you do know that the Smithsonian is not one, but several museums. Each will appeal to a different interest. Also, if not too hot and you are physically fit, I recommend walking from a central downtown hotel rather than driving. Metro is fairly easy to use and any hotel within walking distance of a metro station would also work, but be careful of the areas, choose safe rather than cheap. What you save in a distant hotel will probably end up costing as much by the time you either pay to park or to take Metro everyday.
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