I might change my plans for indoor activities, like going to a museum. I love art, but I'd be more interested in learning a bit about the history of the city, since I know virtually nothing about it.
In which case you probably won't find better than the Museum of London. I've not been to the main branch, only to the Docklands one. Docklands was good and the main branch is probably even better.
One good thing about the UK is that many museums have free admission.
Googling for 'london museums' will also throw up a long list of results.