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 V and A houses a public collection of prints and drawings of every kind imagineable from old master drawings to photographs, playing cards, fashion plates and wallpapers. You have to select what you want to see from the collection http://collections.vam.ac.uk/and book in advance. The behind-the-scenes route up to the study room of v and a is a treat alone (pictured)
2 Willow Road designed by Ernő Goldfinger and completed in 1939. It has been managed by the National Trust and is open to the public. Goldfinger lived here with his wife and their children until his death in 1987.
I only made my first visit to Troy Bar fairly recently but it's already become a Friday night regular, and I feel like I've wasted a lot of nights not being here. Caribbean food, 4 Red Stripe for a tenner, an incredible house jazz band and as good an atmosphere as you'll find in the city.