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.
The thing I love most about London is the juxtaposition of urban and rural, and nowhere is this better illustrated than in Lea Valley Park. A genuine wilderness where I’ve seen everything from ponies, to a rogue Costco outlet, and all less than 20 minutes from my front door.
SOAS university has a small gallery space dedicated to exhibitions about Asia, Africa and the Middle East. There's a small but beautiful Japanese zen garden on the rooftop (currently closed for refurbishment)