When you plan to visit will still be during school holidays, most schools return early in September so London will be very busy. I would definitely buy tickets in advance to guarantee being able to visit, we pre-booked tickets and still had to queue for a short time last year. The Churchill War Rooms are worth visiting and take up to a couple of hours to explore. As JB said, there is plenty to do nearby, adding to his list there is also Buckingham Palace and Trafalgar Square.