Korean? Annandale, although you will need a car to get there.
Yes, first thing that popped into my mind was Honey Pig.
DC is south of the Mason-Dixon line, so there's a lot with a Southern flavor - and a LOT of international fare.
Ollie's Trolley for a local feel. Nando's for Portuguese.
I think my favorite "local flavor" is to go to Eastern Market to walk around the vendors on a Sunday morning and eat - crepes outside or (better still) eat breakfast or lunch inside: blue(berry) buck(wheat) pancakes, the brick, fried green tomatoes, lunches are great too. Lines are long and you'll be rushed once you get to the front for both, but IMO, worth the wait.
Try Native American food at Mitsitam (sp?) Cafe in the new Museum of the American Indian.
If you want to get a Capitol Hill experience, I think I remember that the best places are Charlie Palmers (steakhouse), The Palm and Capital Grille were places that I saw a lot of Congressmen and lobbyists.
I don't know if Washington State has Krispy Kremes, there's one in Alexandria, if it's still there.
It's been so long since I was there regularly that I don't want to recommend any of my favorite holes in the walls in case they're gone or just gone bad :/
Have a fun trip!