I was over there earlier this year. You can easily spend a day just wandering around the various memorials.
Overall the highlight for me was the Al Weiwei exhibition at the The Hirshhorn, but that may have finished by now.
We had prearranged entry tickets to the Capitol Building, and managed to get entry passes in the House of Representatives. Which was all very interesting. Someone arranged those for me, but imagine they can be sorted out online.
The National Archives Building is worth a quick visit to look at some of old American documents.
I think by June it should be fairly warm. While I was there in January the temperature varied from about minus 10 to 15 degrees. We also had freezing rain one day, which meant everything shut for the morning.