You will never be left wanting in London, with something nerdy always around the corner or in the neighboring villages. While you should definitely visit some of the more 'typical' tourist attractions, there are many off the beaten path and for free. If you want to channel Peter Pan go for a stroll through Hyde Park and try to find the locations mentioned in the play. Try and also find the tribute statue, and without a map for a little challenge.
Not far from King's Cross you can visit the filming location for BBC's Sherlock and stop by the British Library. The library boasts an impressive private collection of old manuscripts, including sketches from Da Vinci and one of the few surviving folios of Shakespeare's plays.
A short train ride from central London is the Harry Potter Studio Tour, where you can explore the sound stages used for the films and see many props. You can even have some butter beer, which is the same as in the Wizarding World of Harry Potter in Orlando and Hollywood.