Well I'm not a Londoner but I've taken loads of kids of around that age to London (including the annual Year 7 trips I used to organise when I was teaching), so I'll offer a few suggestions:
1. At the Natural History Museum be VERY selective about what you go to see. Kids love the moving dinosaurs and exhibits such as the (dead) giant beetles can be popular but kids can quickly get bored with a lot of what's there. Do a bit of advance planning:
http://www.nhm.ac.uk/.../galleries/index.html
2. The Science Museum, next door to the NMH, is about a zillion times more interesting for young people. I've often had to drag kids into there (because they though that science was boring) but then had far more trouble dragging them out again over five hours later.
Once again though, it's wise to be selective. Launchpad (which is full of interactive exhibits) is an absolutely essential part of any visit. The aviation and space sections are always popular as well. (Real aircraft hanging from the ceiling, giant jet engines to be photographed alongside, a genuine crew module from a spaceship, complete with the burn marks from re-entry into the Earth's atmosphere, etc, etc).
Unfortunately HMS Belfast (which would normally feature high on my list of priorities) is currently closed, so I'll have to move on to . . .
3. The Monument:
It's a great testy of everyone's fitness, climbing to the top, with loads of history (and some great views) thrown in:
http://www.themonument.info/
4. The Imperial War Museum also benefits well from advance planning if you're to be able to skip the boring bits:
http://www.iwm.org.uk/visits/iwm-london
5. If the weather is good (or, at least, dry) the street entertainers in Covent Garden provide some really good free entertainment:
http://www.coventgard...eet-performers/page/1
6. A trip on the river is fun but kids can get bored (and cold) quite quickly so it's best to keep it short. Don't bother with the guided tours. (Kids couldn't care less about which architect - who they've never heard of - designed a building for which dignitary - who they've also never heard of). Just use the regular river bus services. The short trip (20 minutes) between Embankment Pier and Tower Pier takes in a lot of the sights:
http://www.tfl.gov.uk...rsenal-embankment.pdf
but you might want to consider staying on until . . .
7. Greenwich (with the Cutty Sark and the National Maritime Museum):
http://www.visitgreen...to-visit-in-greenwich
Chris