Not the most exciting place inside, but a lovely rooftop area which not all that many people seem to know about, 2 mins walk from Old St station. If it's packed then The Red Lion is just up the road.
Key community arts centre, near to Shoreditch High St station. The cinema is handy and the music programming is solid, but they also host a variety of other interesting nights, most notably Jawdance, London’s (arguably) best spoken word night
The Greenwich Observatory with its green laser beam can transfigure any night sky. If you decide to cross the Thames via the foot tunnel, make sure you look out for the green ray. You can be on one side of the meridian in a minute, and cross to the other side on the next, without even realising it.