As someone who spends quite a bit of time in downtown DC, I got excited by seeing your post, since I certainly have some ideas for you.
Starting with sites to see: in response to the previous post debating the spy museum, it is a great destination for people that are intrigued by the intelligence agencies and spy scene of DC and that sort of thing in general. To fully enjoy the museum and get your money's worth it is worth devoting a half day to exploring it all. That said, I would not make this the priority if you have not seen the gems that happen to be free in DC. A lot of it depends on what you like. I personally love the art museums (The National Gallery has an incredible collection of renowned masterpieces studied the world over, and here you get to see them for free! The volunteers at the information desk are wonderful and if you tell them the sort of thing you like they can guide you towards the best route through the museum for your taste, since otherwise it can be overwhelming. The National Portrait Gallery and Sculpture Garden are also very good). The Natural History Museum, Air & Space, and History museum are all awesome if you have not seen them yet, great for kids, but time it right (summer weekends early afternoon are the most touristy times).
Personally, my favorite place in DC in terms of architecture is the Thomas Jefferson building of the Library of Congress; it never fails to leave me in awe as I gaze up through the arches and intricate murals, reading the quotes all over tat come from the great minds that built our country. Another benefit is that it's generally not as touristy as other places in DC in July.
Now for the dining: The eatery in the Native American museum actually has many delicious, unique items labelled gluten free! It is a beautiful museum to pop into anyway and doesn't take too long. As for restaurants, these are the ones I have had success in, around the downtown area:
-Zaytinya: one of my favorite places in DC to eat. Don't go if you are too too hungry as they serve tapas, but they have a dedicated gluten free menu that includes a large part of their regular menu, so you get a ton of creatively crafted options.
-Clyde's in Chinatown: their seafood salad or other salads are substantial, and you can get their crab&artichoke dip with cucumber slices instead of bread.
- Le Pan Quotidian in Georgetown: has a gluten-free quiche, good gazpacho, and good salads. However, its also is a bakery so you do have to endure seeing that sort of thing all over
- Paparazzi in Georgetown: a nice Italian restaurant in a good location, with gluten-free pasta to boot
- Hello Cupcake in Dupont Circle: has gluten-free cupcakes
- Heritage of India in Dupont Circle: excellent indian food that is naturally gluten-free, the specials are very reasonable prices and the sunday brunch is a steal
- Raku in Bethesda and Dupont Circle: yummy asian fusion with outdoor dining and good gluten-free offerings. They have a gluten-free menu, but for some reason some of the servers do not know about it so you have to try and get an informed person to talk to. The food is great though!
Hope this helps!