this is the absolute fastest way to get there from mechanicsville. I've done it a bunch.
take rte 5 up to suitland parkway, then go left and ride that into the city where it becomes south capitol street. you're only on south capitol a short time, when you're passing the best western hotel on the left you need to stay in the middle lane for the downtown ramp that takes you up to the left. I believe the sign says "downtown" or "D st." or both. you'll be able to see the ramp you're getting on, just stay in the middle and you'll see where it goes up and then curves to the left. this will put you on 295 headed west. you go down 295 and take the 12th street exit. it's clearly marked. when you're going up 12th street you want to get over to 14th and keep heading north. theres a number of lefts you can take and you've got plenty of time to do so. I usually go left on pennsylvania ave then go right on 14th. you ride 14th several blocks all the way up to P street and make a left on that, then go several blocks on P street, around dupont circle, keep going, eventually you'll pop out on wisconsin ave and you're almost there. make a right on wisconsin, and you can go up and make a left on reservoirrd to take you to the hospital or you can do this: make the quicker left on Q street, then right on 35th, then left on reservoir rd.
I have to take someone to a dr's appointment now & when I get back I'll tell you how to come home because it is slightly different.