Where exactly are you staying? What I consider to be "Midtown" is Metro Center. That's a very, very long walk across town to the Kennedy Center. Take the Blue/Orange Metro line to Foggy Bottom. Cabs are not the best option. Not too expensive, but hard to track down.
Amtrak is hit or miss, IMO. Sometimes you find $49 trains when you buy months in advance. More often, it's $80-$120, each way. Trains are 3-4 hours, bus is 4.5 hours. I'm up and back from New York on Bolt Bus probably once a month and only on one occasion have I encountered traffic so bad that I felt it would have been worth the extra money to take the train.
CapnHook- Take Bolt/Megabus to the 10th & H St. stop. Metro Center is 2 blocks away and you can take the Blue/Orange line to Foggy Bottom, as others have mentioned. It's just 3 stops from Metro Center.
Like a firework unexploded
Wanting life but never
knowing how