It might seem a very obvious and touristy place, but Covent Garden is worth a look. We've been there twice in the last two years and,certainly this year, the quality of goods on sale in the market and the street performers were far better than I expected. i even bought some clothes for a wedding we're going to from a shop just off Southampton Row in Holborn for considerably less than I could have got them in Manchester.
The Imperial War Museum as doggybob said is well worth a visit, allow half a day. The Ace Cafe as dave mentioned is also worth dropping into even if you only stop for a coffee and a bacon roll. I went there a few years ago on my motorbike on a week day and while I was trying to find a good spot to take a photo with the cafe in the background a man wandered up and made a few suggestions, even helping me manhandle the beast into position for a good picture. It turned out he owned the place.
We've eaten quite a few times in the White Lion and found the food to be very good quality and prices really quite reasonable especially considering where it is.
http://www.coventgardenlondonuk.com/pubs/white-lion-pub
Maybe it's because I live 200 miles away as well as the fact that my mother's family all hail from the East End that I enjoy going there, but we'll be going back next year as well for a visit. In fairness we never "do the tourist thing" wherever we go; London, Paris, Rome, New York or Venice, much preferring to be part of the area and not just a visitor. There are plenty of good pubs and loads of shops that are not aimed at the tourists and just to walk past the Royal Courts of Justice, sit outside Somerset House and have a coffee or do the same in Simpson's in the Strand is all part of the experience.
http://www.simpsonsinthestrand.co.uk/
Last year we went to Kensington High Street to try and find where Mrs. Beard used to work in the '70s. Although Biba has long gone, there is still a department store there with a restaurant on the top floor and a roof garden which is now a branch of the Virgin empire. Pretty expensive but for the price of a cocktail or two you can spend time with the penguins and "cows" and get a view that takes in most of London from Battersea Power Station to Canary Wharf.
http://www.virginlimitededition.com/en/the-roof-gardens
London, like most places, is what you make it. Whereas some European cities are stuck very much in previous centuries and others have been re-built after bombing, in a similar way to how they were beforehand, London has replaced some old buildings with new, and in some cases, very new structures which can make it look a bit mixed up from some viewpoints, but it doesn't have the graffiti of Rome, the rudeness of Paris or the stag parties of Amsterdam, Prague or Hamburg.