I have a 7yo daughter who LOVES the DC museums. I think your daughter would too.
We lived in DC for 8 years and moved to AZ last summer, when my kids were 6.5yo, almost 4yo, and 5 months. The older 2 always enjoyed the museums and remember a lot of the stuff we did a few years back. My 4yo told his new teacher in AZ all about the frog exhibit we went to at
National Geographic when he was less than 3yo. She is 35 and didn't even know what "National Geographic" was! (Well, obviously she has heard of it, but I mean she had no idea they had a physical location and thought he meant online, then asked me if it was a museum or something.) So I am super happy that my kids have gotten to experience all the DC stuff that they have. They both LOVE the Air and Space Museums, National History, Building Museum, American History, the Zoo (though I admit it isn't so great, but the kids like it just fine and it's free). Insect Zoo and butterfly exhibit in the Natural History Museum are very good with kids, as is the Spark! Lab in the American History Museum. Washington Monument, Dept of Treasury, Memorials, etc. are all neat for kids too. I've done the Holocaust Museum twice with my husband, but haven't taken the kids. I am sure they would like the kids exhibit there and even some of the permanent exhibit, but I wanted to be able to read stuff and spend a lot of time there so that's why we went alone.
And where are we right now? Spending 2 weeks in DC! Yes, even in the July swamp-like conditions it's a fun place to be. (Now, we live in southern AZ, so that's insanely hot now to the point that DC feels cool, but still!) But when you're coming, late Sept/early Oct, it will be even better. Way less hot and humid, and way less crowded.
The Omni is a decent place to stay and there is a great restaurant/coffee shop across the street from it, Open City. I highly recommend eating there, it's cheap and the food is good. Also, the Washington Hilton on Connecticut/19th is nice and usually pretty cheap. Right now I am staying at the Georgetown Suites which can have amazing deals, plus you get a full kitchen, free breakfast, and separate bedroom. I'm paying $110/night for an 800+sqft 1 bedroom unit. The circulator bus picks up right outside basically and it's only $1. And so much in walking distance in Georgetown, including both Sprinkles and Georgetown Cupcake

.
We are big time WDW people but our last trip was in November 2009 and my kids actually asked to come back to DC (granted they have friends here) rather than WDW.