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Cosi
This particular location of the coffee and sandwich shop offers free WiFi.
Freedom Plaza and Pershing Park
The Open Park Project provides free WiFi for both Freedom Plaza and Pershing Park.
Kogod Courtyard
Free WiFi is available in the Kogod Courtyard at the Reynold Center Smithsonian American Art Museum and the National Portrait Gallery. There is a small cafe in the courtyard where you get coffee and sandwiches. The courtyard is a peaceful, quiet place to escape for a break. Note that your bag will be inspected when you come into the museum.
Martin Luther King Jr. Memorial Library
D.C.'s central public library is enormous, and, designed by modern legend Mies van der Rohe, is one of the best examples of modern architecture in the city. Sadly, it has seen better days, and isn't as lovely a place to hang out as it should be. It's still quite the landmark, though, and it has a really nice bookstore inside. As with all libraries, free WiFi & public terminals.