Getting a hotel near the museum isn't necessary. Smithsonian and Federal Triangle Stations are both an easy walk to the museum. The Metro is easy, clean, fast and reasonably priced.
Just be prepared to spend an arm and a leg and possibly your first born, especially this time of year!
The Gaylord is nice.
There are quite a few hotels North of the Mall around the Metro Center and Gallery Place stops. If you won't have a car, you are not limited to staying in that area, however. Anywhere near a Metro stop will provide easy transport to the Mall, and you may find cheaper lodging in Arlington or other surrounding areas.
There are many hotels near the African American Museum. Look on the internet to see them all. I stay at the Washington Hilton whenever I visit the museum and take a taxi from the hotel and back to the hotel.
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