What we did
We had did a similar vacation a few years ago with our 3 teenage boys. We stayed at the Embassy Suites outside of DC but right on the subway line. It was great because it wasn't nearly as expensive as staying right in DC & the subway ride was short. We enjoyed the huge 'free' breakfast before we left the hotel for the day. Since we had a bigger breakfast, we ate a smaller lunch later in the day. When we got back to the hotel, we enjoyed the 'free' happy hour & snacks. The kids had no interest in going out, so they ordered a pizza & hung out at the hotel & my husband & I walked to the nearby restaurants. We never used our car until the day we were leaving & we went to Arlington and Alexandria.
We were gone 10 days on that trip. We went to Williamsburg, Norfolk (great tour of the largest Navy base in the world), Virginia Beach, Annapolis (toured the Naval Academy) and the sights of DC. It was an exhausting trip, but one of my very favorites!
Have fun!