I don't know Paris, but I live in greater London (suburbs, not central), so I know parts of London OK.http://ladydinahs.com/
is somewhere I want to go. (It's not specifically a "tourist" place, but hey.)
Definitely see a show in the West End, and/or The Globe. Otherwise, The Old Vic is also a great theatre that puts on wonderful performances, and isn't too expensive - I highly recommend it.
The Royal Albert Hall usually has something good on, too. The very cheapest tickets are under £10 for some things, and you can buy standing tickets on the night.
I'm pretty sure St Paul's cathedral charges to get in.
If you want to see Westminster Abbey and Westminster Cathedral (2 different buildings), exit London Victoria train station (overground) at the main exit out front, then turn right and walk along the main road. (Huge department stores etc along there, kind of hard to miss.) This road will also take you to the Houses of Parliament/Big Ben, and then across the bridge to the London Eye. When you exit Victoria Station, there are usually specifically hired people standing outside to give directions/answer questions (or at least, they're always there when I go).
You can watch the changing of the guards at Buckingham Palace for free. Just show up in front of the palace at the right time, and enjoy a free show of pomp and ceremony. http://changing-guard.com/dates-times/buckingham-palace.html
If you encounter the guards standing outside the entrances/exits on guard duty, please don't poke/prod/hug/kiss/drape oneself all over them in an effort to get them to move! (Not that I think your daughter necessarily would - it's just something that not everyone seems to realise.)
to be stood stone-statue-still on that duty, and not react to passers-by.
If using London Underground (the Tube), you can find a Tube map on the internet; if possible, I'd recommend printing one out before going. (Colour is best!) Or you can buy one from loads of places. Every time I get the Tube in London, everyone is crowded around the Tube maps on the wall to try and figure out where they're going. If you have your own Tube map, you don't have to crowd there with them. (I've lived in greater London for most of my life, been into central countless times, and have still resorted to buying maps.)
Ditto poundcake's point about not stopping in the way on the Tube. My method if I think I'm going the wrong way, is to find a pillar or something to stand behind, or somewhere to the side out the way.
Also on the Tube - head to the end of the platform (especially the front end, in my experience, or whichever is furthest away from the platform entrance) for train carriages that aren't always as busy. (Note, this isn't fool-proof, and can depend which direction you are travelling, what time of day it is, etc etc.) Most people are rushing/can't be bothered, and it always amuses me when I get the tube from London Victoria to London Paddington, to walk down to the front end of the platform, get in the front carriage, sit down in the carriage where it's only half full, then look towards the middle of the train, where people are stood squashed together like sardines.
Public conveniences at major train stations (such as London Victoria) cost about 30 pence to use, although it can vary. (They usually have a turnstile.) Keep some change just in case you need it!
If your daughter is into punk/goth/alternative sort of stuff, Camden Market (and the surrounding area) can be a good place. I rather like it, although it can be a bit samey with a lot of the shops/stalls selling similar stuff, and can be expensive.
If she likes vintage, Brick Lane is the place to go.
Some people sell laughing gas/helium highs on the street; although this is *legal*, it's not really recommended. (IMO.) There have been problems with so-called "legal highs" (ie, drugs sold for recreational use that haven't been banned yet). I'm sure your daughter is sensible, but I thought it wise to mention just in case.