The Barbican is my favourite building or place in London. One day I will have enough money to buy one of those apartments. It really is an amazing piece of Brutalism design. No doubt this will be on your to-do list, but what might not be is the Architecture Tour. It's about £12 for a 90 minute tour of the area and its design history. I did it last year and it was brilliant. Just don't do it mid-winter, it's like a wind tunnel round there and bloody freezing.
Barbican, London EC2Y, UK