Maybe an obvious choice, but the Barbican is always good. My favourite space in there is the curve gallery, last time I went was for Cory Arcangel's 'Beat the Champ'. I love the architecture, although not many people do. I think it was voted 'London's ugliest building' in 2003.
It's really close to my house and one of my favourite pubs in London! (I even went before it was close to where I lived!) It is London's first cooperatively owned pubs and has lovely events and workshops, exercise classes and great parties, and good beer! It's lovely in the beer garden in the hot summer and equally great in the winter, next to a real fire.
London is so vast I am forever coming across new hidden gems. Walking around the streets and getting lost is the best way to find London's best kept secrets.