Downtown DC:
I'm staying with the nicer hotels, because you mentioned you wanted to impress her. Let me know if these are too expensive.
Mandarin Oriental. We stayed here for a weekend away (got it on a special deal, so the price wasn't insane...just slightly shocking), and it's beautiful. 2 block walk to the Smithsonian Museums...though it's situated in a wierd area chock full of government buildings. Overlooks the river, though.
The Willard. A DC landmark, and impressive as hell, especially the entry way, which looks like it was lifted straight from an old-school European palace imho.
Neither of the two above would I do unless you find a special deal (check Orbitz or whatever), they're just too expensive otherwise imho (standard rates are like 400 a night, no joke). But both are amazing...
Grand Hyatt, DC. It's a conference center type thing, pretty nice, but easy access to Metro.
DuPont: Further from the center of town (though Metro access and a very short ride to downtown), but nice area with much better night life, and tends to be less expensive.
Hotel Palomar. Modern and nice. Haven't stayed there, but have heard good things about it.
Park Hyatt. It's a nice hotel, but has the benefit of having a really excellent restaurant in-house.
Blue Duck Tavern. I eat at a lot of the good places near Downtown (thanks to my job, these meals are on my firm), but this was really a great place to eat. Fantastic.
For dining options, some off the top of my head that I've been to near Downtown (google them to read about them):
Best in Town: Citronelle. Not even close. It's also atrociously expensive, but this restaurant is world-quality.
Expensive, but good: Source, Central, CityZen, Tosca, Kinkead's, 1789 (but if you're doing French, just bite the bullet and look at Citronelle)
Other ones I can think of : Palena (highly recommended, a little outside downtown), TenPenh, Oyamel, Cafe Atlantico, Jaleo, Ceiba, Oya.
Note: All of these restaurants are expensive, but food in DC tends to be expensive anyways. I routinely spend 10 bucks on lunch, just for a freaking sandwich and drink from Cosi. I've eaten at all of these restaurants, and I think they're all good...some better than others, but they span a huge range of food choices.
If you let me know what you're interested in (American, etc.) I can provide other options. That is, if you want a nice fat steak, I'd go for BLT Steak or Capital Grill, but I wouldn't mention them otherwise. If you want a classic, check Old Ebbitt's Grille, where you'll likely see a Senator or two.
Frommer's has a decent DC guide to restaurants
that you may want to look at.