I've never stayed at any of the hotels right near any of the museums so I can't give you exact info. However, I can tell you that if you can't find one within in your budget in the city itself, you can probably find one in one of the smaller burbs. Not wanting (or able to) drive in the city would be no problem. When you check out hotel listings, see if they mention how close they are to one of the Metro stations (or call and ask). The subway will get you close enough to the National Gallery of Art you can walk from the station. The last time I was on the Metro, it was still one of the cleanest and safest ways to get around the city without a car. Just a few dollars will get you almost anywhere (unless they've really hiked up the prices).