Last time I did The Blitz walk, which was awesome. I see they have many more walks now. Hampton court is quite worthwhile; take the train, it's so easy. The war rooms were good, but pick a time when they're not going to be crowded, like opening on a weekday. The Tower! There's a good tour of the Old Globe. The Victoria & Albert museum is good, not as crowded as the British, has a good cafe, and is around the corner from Allen House.
About 150 years ago, the British went around the Mediterranean making plaster casts of zillions of statues, busts, engravings, etc. They brought them home and put them indoors in the V&A. 150 years later, pollution has really degraded the originals, which still are mostly outdoors, but the ones in the museum look great.
If you want to spend another week in another location, here's what we did. Pick a rail center or crossroads and stay in a hotel near the station. Get the appropriate railpass and take day trips. We chose Reading, only 40 minutes from London, got the London Plus pass, and went to Bath and several other places, some chosen at random. You could easily go to Stonehenge or Avebury (bigger, I think better, less famous), Southampton, Bristol, Penzance, Oxford, even Wales from here. (If your destination is outside the pass zone, you just pay the difference.) Buses are good too.