Ordinarily the Monument would be a must-visit: not only is it an important piece of London’s history but you get a superb view from the top. But it’s closed for the rest of the year for renovation.
If you want to catch a play, try Shakespeare’s Globe.
Definitely go to the British Museum. It’s amazing. I was there several hours just seeing the Egyptian and Mesopotamian areas, bu there’s also Greco-Roman, Oriental, etc. And they have all the famous everything (or nearly so) so if you have any interest in that sort of thing at all you should stumble across hugely important artifacts. With some work you can walk from there through a theatre district and get to the National Gallery/Trafalgar Square. The Tube is great, but actually seeing all the neat things above ground is good too.
If you are at all interested in Jack the Ripper you can take a tour in the East End, maybe see the London Hospital Museum, and there’s a new Ripper exhibit opening in the Museum in Docklands very soon that runs until nearly the end of the year.
The Victoria & Albert Museum is good too, and within walking distance/quick Tube shot of Hyde Park for a combo trip.
Westminster Cathedral (i.e. not the Abbey) is worth a look for its Byzantine architectural style.