You are staying in a weird part of town in terms of eateries. SW shuts down at night because it’s pretty much all govt offices and sightseeing locations. But the metro system is easy, and you’re not far from Capitol Hill.
Others may rag on this place, but a great breakfast dive and great deal is at Pete’s on 2nd St SE. Right by the Bank of America, near the corner of Independence and 2nd. If you’re doing morning sightseeing on the Hill, it’s a great location, excellent service, pretty much cheap as dirt, and good eggs/scrapple/decent grits. At lunchtime you’ll probably run into a lot of Hill staffers, who, while obnoxious, can make for decent sightseeing in their own right.
Matchbox is definitely a good bet. A lot of chains in the downtown area. You’re not too far from Naan and Beyond, which serves Indian sandwiches on naan. And, of course, if you really need a good burger fix, there’s always Five Guys.