I'm staying at the same hotel in a couple of weeks and while I haven't tried anything yet, I am the queen of research and a good deal. This is what I have found right by the Hilton Bistro du Coin and Teaism. Let me know how it goes. Also check the HIlton website they always list the distance to close restaurants.
Olives is right across the street from the hotel. Italian food and loud - 2 8 yr olds won't be noticed! There are a bunch of sandwich places around there for the office crowd, but it's pretty upscale other than that.
If you'll be heading to the museums, Matchbox is perfect - be sure to make a reservation - it's always crowded. Ella's pizza is also great - also in Chinatown, near the Spy Museum if you're heading there.
Well, no one else has answered you, so I'll give it a stab. Not having a car limits you, and the kids probably don't help. So, I'm going to send you to a pretty basic option - Old Ebbitt. If you were sans kids, we could definitely up the ante. But at Ebbitt, you can do pretty well with their raw bar (awesome shrimp cocktail) and crabcakes, and the kids will have a lot of options. It's loud enough that you don't need to worry about the kids causing a ruckus, but it's also quite pretty (just make sure they don't sit you in the office building atrium. I particularly like the bar area to the left when you walk in the door. It's cozier.
A bit further afield would be Chef Geoff's, which again, isn't the best restaurant ever, but I think the kids could enjoy their pizza or burgers.
The good thing will be your walk there - go down 16th, toward the White House, and across Lafayette Park. It's beautiful (and safe, don't worry) at night. Take a left when you get to the White House, then a right at the end of the block, and you'll be a block from Ebbitt.
If the kids are particularly adventurous, then we could come up with some super options.