Canary Wharf ice rink - get the DLR there and sit at the front - v futuristic top view
Round that way is also climbing the Millennium dome
And Greenwich Maritime museum
cutty sark
If you are going to the shard and they have any aspirations to medicine or nursing then the old london operating theatre is good (they might be a bit young) - and if you are there Friday or Sat then borough market great to eat in
around London Bridge tower bridge HMS Belfast and the london dungeon (and tower bridge, obvs)
Boat along the Thames (there are speeds ones)
Tower of London - probably only if they are really into horrible history (if they are there are hh boat tours)
China town for lunch or dinner - all twwens and young teens seem to love it
Transport museum - I think of that more for younger kids but if they are into that. Churchill war rooms - maybe they are a bit young but it’s ace if they like WW2 stuff
If they are at the shopping stage then Camden
Jules Verne War of the world experience
Open top bus tour
Science and natural history museums
Harry Potter tour, of course
300 steps up monument if they like heights and you need to tire them out