This may not be much of a secret but deserves a mention. I find it hard to go anywhere near this place without having to go there. Freshly baked Beigels hot salt beef and it never closes, 24 hours a day 7 days a week.
The Barbican is worth visiting for it's brutalist architecture alone. The concert programming is really diverse; it's home to the BBC Syphony Orchestra, but you'll also find bands like the Dirty Projectors and the National amongst the classical and contemporary composer line up. Plenty of art, dance and theatre too, however, the Conservatory is the real gem that's not to be missed.
There are many quirky book stores in London, but I really love the selection that Daunt Books have, they often have bespoke editions which you won't find in other book stores. Also, the people that work there are so helpful and knowledgable. I can never stop myself from walking in!