Food!
Not that I've eaten at either of these places, its been a while since I was in London, but I would check out St. John's Restaurant, owned and operated by Fergus Henderson, one of the primary chefs responsible for the resurgence of the nose to tail philosophy of cooking. Essentially he focuses on offal and other neglected cuts of meat. He's a super cool guy and while the restaurant is kind of expensive it sounds like a treat.
https://www.stjohngroup.uk.com/
Or you could check out Duck and Waffle Local. Recently opened, it is a more affordable take on the Duck and Waffle luxury restaurant located on the top floor of a sky scraper. They do (relatively) affordable takes on their popular dishes in a pub setting.
http://www.duckandwafflelocal.com/
Again I've been to neither and while I'm sure there are 40 thousand different restaurants, I think these two would be on my list to visit.
Other than that, an obvious suggestion is to go visit the free and world class British Museum. Total no brainer.