I work in DC. It's a beautiful city and would take a several weeks to see even the major attractions.
I think my advice would be to get a hotel in Alexandria. Unless your are planning to do night things like the theatre or sports (Wizards, Capitals, Nationals) having a daily retreat to a nice place in Alexandria would be healing. Alternate locations could be Crystal City or Roslynn.
It's best to not attempt to navigate your vehicle around the city. It can be very daunting with parking, one way streets, taxi drivers...
With your hiking skills, you will easily be able to walk the entire length of the mall. I would start at the Lincoln Memorial, and see the Vietnam War Memorial and Korean War memorial. If you want to take a longer walk, head over to the Jefferson Memorial passing the FDR Memorial. Head toward the Washington Monument from there and check out the Mint store on the right. If you want to tour the Mint, you go around to the other side. Next to the Mint (on 14th street) is the Holocaust Museum. Both of those places typically has timed cards they hand out, so you can't always just walk in for a tour. The Mint takes about 30 minutes. The Holocaust Museum will take at least an hour even if you have a fast pace run through. It is a moving experience.
If you skipped those two places and are still headed toward the Washington Monument, bear to your left and if you look closely, you'll find the WWI War Memorial (which needs restoring). Go ahead and check out the Washington Monument then head back toward the Lincoln Memorial. You will see the "New" WWII War Memorial in the center of the Mall just before the reflection pool. You have invested about 6 hours by this time. I would suggest that you then walk past the Lincoln Memorial and cross the Memorial Bridge to the Arlington National Cemetery. JFK Flame, Changing of the Guard, Space Shuttle Memorial and way too many white stones.
There are not many food establishments on this walking tour, so keep your eyes open for refreshment stops.
The next day, I would go to the other end of the mall and find the Supreme Court. It only takes a few minutes to look around and perhaps get a tour if they are not in session. The Capitol Building typically requires a pass from your Congressman or Senator to see the full galleys (Not sure, its been a while since I've been there and security changes periodically). From the Capitol looking toward the Washington Monument, you will have the Senate office building on your right and the Congress offices on your right. You don't usually require a pass to enter these buildings, but the security is pretty tight. You can actually go to your representatives office if you like and maybe they will be in and say hello. I probably would just skip that. Heading down the left side of the mall, you'll see the National Arboretum and then the Smithsonium National Air and Space Museum (must see). After that the old Smithsonium Museum (Castle) is nice with a pretty flower garden on the back side. (You are now by a Metro Stop, so you could start here and go backwards). After that, go across the mall to the other side. You probably would enjoy The Smithsoneum Natural History, American History, and ending up at the National Archives Building. Thats probably it for day two.
I'm out of time just now. If you like this, let me know and I'll jot down some more notes.
