Was in London last week and had a blast. First time on London Eye, took a boat ride, and went pub crawling around the West End and Camden.
Speaking of Camden, you mentioned plays - there was a Camden comedy fringe festival on last week... most of the shows are short, we ended up at three (!), names from memory:
The Dave and Dave Show
Not the Weather Report
For Your Viewing Pleasure - this one was totally hilarious and had the added advantage that the theatre it took place in was attached to a pub (although sadly the pub was a dump).
Speakers corner always good for a laugh too. Check online for events.