If you want a peaceful day out of the busy city, Highgate doesn't really feel like London. The cemetery is interesting and peaceful, there's a tour of the old half which is definitely worth the money. Karl Marx, among others are buried there.
Coming here as a child and even now the collection is utterly astounding I can spend hours in the fossel and mineral room. Plus the architecture of the building is completely unique and breath taking.
Free natural ponds, open all year round (check online for varying opening hours according to seasons). Get an absolute get away feeling right in the heart of London, a treat if you enjoy swimming with ducks. Amazing park to enjoy a few games of cards and a walk around afterward too...
This independent cinema near Angel in North London is over 100 years old and has had several lives including hosting early Sex Pistol gigs. It now serves as my favourite place to watch films (along with the Rio in Dalson).