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.
The closest you can get to Little Italy in the east.
The owners are amazing, the atmosphere is relaxed, the lunches are abundant and homemade.
The cakes are amazing, try Torta Clara (pear and chocolate) for the ultimate bliss.
Aside from the excellent cultural facility's the Barbican Centre has to offer, ranging from theatre, art, music and dance. The space outside the Centre holds personal value in its ability to evoke nostalgia of the Emerald City, in the slightly disturbing film 'Return to Oz'. I enjoy residing by the fountains, where I draw, read books and feed ducks.