11 hour layover in London... what should I do? and how to get around?

Cheapest and simplest option is to buy an all zones travel-card for the underground. This will get you into central London from the airport and will give you unlimited use of the underground for the rest of the day. Underground trains are very frequent in central London. Last time I bought one the cost was �7 but i suspect they have gone up since. No doubt another poster has the current cost.