that's the old post office pavilion. right now it's mostly just a food court with some shops. i think it might be 2 levels worth? but the selling point is the bell tower observation deck. the pic before that one of the washington monument was taken from there, although most of those smudges are from the plastic 'window,' not the camera. most of the window area is actually open but they have firm wires in place that you can't really get an SLR lens through unfortunately (which makes me appreciate the eiffel tower at the paris in vegas even more since they built in camera holes).
it's considered the poor man's washington monument because it doesn't require a ticket, there's never a line and it does offer some interesting views of the city. you ride up in a glass elevator that overlooks inside the building (although my shot was actually through a window on the 10th floor (the observation deck is on the 12th)).
i'm sure you'll find plenty to shoot, especially with the cherry trees. DC is loaded with monuments of all kinds, to people and events. my top 2 tips would be: (1) get your washington monument tickets in advance through nps.gov - it costs $1.50 per ticket, but it's so worth it. unless you line up really early in the morning, you won't get a ticket otherwise. (2) contact your senator's office well in advance. last time i checked, white house tours were offered on groups of 10 or more so they may have something they can arrange, but it has to be done at least 3-6 months in advance. also your senator may be able to arrange for a special tour of the capitol. a page from their office will lead it starting at the senator's office and you get to take the underground tram the senators take to get to the capitol.
you'd also do yourself a great service by taking a tour or two. i found them so overwhelmingly informative that i've convinced myself i need to take one of them here in NYC. i did the DC Ducks tour which is an amphibious vehicle that drives around the streets of DC pointing stuff out and then ends up in the patomac. one of the things i learned that made me want to go back and shoot it was the sculpture behind the holocaust museum when i learned it was meant to be a dismantled swastika. unfortunately i had already visited the museum prior to the tour and never was able to get back there, not even this trip. there's so many things going on in dc, a lot of symbolism, a lot of overtness, a lot of meaning and a lot of anecdotes.