Last time we were in London, we got tickets for The Shard coupled with a river cruise. We enjoyed both, weather was glorious that day so the views were amazing.
We went to Spitalfields Market, lots of variety of shops/stalls, great indoor "street" food with communal tables, don't bother with the "proper" sit-down restaurants (we tried a few and walked out of them all, they haven't got their act together for service). Some interesting sculpture around the outside.
We've always enjoyed Covent Garden, getting something to eat/drink and watching the buskers.
Londontown.com often has useful information on date specific events.
We've done lots of walking tours of London, just turned up and paid in the past, not sure if they operate differently now. We particularly enjoyed Legal & Illegal London, Jack the Ripper, Oscar Wilde's London, Shakespeare's London.
I know you don't want a show but you might enjoy the cinemas at Leicester Square.