Public libraries
Tenley/Friendship Library
Cleveland Park Library
Palisades Library
Free WiFi and public terminals:
Contact
It's not at all hard to get a free internet connection in the Upper Northwest: Politics & Prose above has free internet, as do a bunch of cafes, and of course all the public libraries.
Firehook Bakery
A great place to check emails, this overpriced bakery has free WiFi and a beautiful hidden courtyard & fountain in the back.
Whole Foods
The giant Whole Foods supermarket just north of the Tenleytown/AU metro station has free WiFi on the second floor.
Epicurean and Co.
A huge, bizarre Chinese buffet, ice cream parlor, sushi bar, convenience store, cafe, full bar, occasional jazz club, and all around popular lounging spot for UDC students and faculty, with free WiFi.
OpenCity
Late night "diner" more of a large cafe with free WiFi, good tea, booze, and vegan friendly to boot!