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.
I grew up in Brighton which is full of cheesy bars like this. If you can handle being surrounded by multiple screens playing Kylie at full volume, its pretty much the cheapest bar in central london. The cliental is so varied its great for people watching and completely un pretentious.
I worked at the Barbican for several years and during that time became very fond of its architecture. One of my favourite spaces within the centre is the conservatory, a giant tropical garden in the middle of a concrete city.