It's a members-only library but you might be granted access with a research request. The writer's room is great but there's a few special, hidden corners among the bookshelves to tuck yourself away in.
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.
Being an East London boy at heart I am partial to an occasional fix of café culture. Pellici's has been running since the start of the last century, to which me, my father and my father's fathers have all enjoyed its mix of tasty Italian/London grub and energetic atmosphere. If you're visiting, make sure you drop by, as it will provide a valuable, forever-depleting education into what East London is really all about.