I spent five lovely, sunny days in London in late September, walking all around Westminster, Piccadilly, Mayfair, and environs. It's taken me a while to process the trip, but here's what I think worth sharing, in terms of discoveries:
1) If you're going to see any show while in London, make an effort to see The Play that Goes Wrong at the Duchess Theatre on Catherine Street. It is hilarious from start to finish. Speaking of the start, be sure to get to the theater 20 minutes early, because the play actually starts before the advertised curtain time. Very unusual, but the pre-play reinforces the theme.
2) The Bank of England on Threadneedle Street is worth a visit for their museum, which has exhibits on the history of currency and on counterfeiting. Assuming you've bothered to pick up any currency, rather than tapping your card everywhere, you can check all your British notes with a special light to see whether they are fake! And if you find a fake... well, there's a gift shop in the next room to fob it off quickly ;-).
3) If you're walking around and need a place to rest your feet for a few minutes, pop into a library. The Westminster reference library is a wonderful place. I found it fortuitously on the Apple map app when looking for someplace to sit after a day of walking, where I wouldn't feel like I needed to buy something to justify taking up space. After all, there are only so many coffees one can drink in a day. Like many libraries these days, they sell used books, cast-offs from their collection. I found a wonderful mid-twentieth century guidebook of the Cotswolds (one of a national series, kind of like the British version of the American WPA guides) for all of one pound. Their shelves hold lots of books on art, architecture and history worth browsing.
Finally, I want to give a shout out to the Rick Steves Travel Forum and all the people who provided helpful tips or asked the right questions. Thanks for helping me plan a successful trip!