If you're going fairly soon and are watching the pennies, just a stroll through one of the parks can be entertaining, watching all the tourists and locals sunbathing in their lunch hour.
There are all sorts of novelty guided tours, such as the ones by the homeless, or instead of actually going to a theatre, maybe you'd like to be shown round behind the scenes.
When I last went to London just for fun with my sister, we visited the Churchill War rooms and went to a musical. I decided that history was more interesting than art.
Some other novelty walking tours (only 10 pounds each) I have found, include a "Jack the Ripper", "Literary London", Beatles-themed, Shakespeare and Dickens-themed, Oscar Wilde-themed, Harry Potter-themed, "Really inside Covent Garden", Sherlock Holmes' London, Undiscovered London, Unexpected London ......
www.walks.com/our-walks/all-walks/