This, regarding safety.
As for things to do, it depends on cost.
The Shard is opening in February (or March), no idea what that's like, but isn't cheap - I'm also not sure how easy it is getting tickets.
Madam Tussauds - Not cheap, but could be interesting
The Planetarium - If she likes space she'll love this.
Museum Wise, I'd say British or Science are the best, unless she particularly likes Dinosaurs, at which point the Natural History would be the one of choice - it might be worth seeing what each has got at the time you go, and base it on what she likes to be fair, as my views a more male perspective.
It might be nice to have a picnic in Hyde Park, if you're going for something nice.
Nothing is a real pain to get to if you use the Tube, and she should be fine on it. In fact, teaching her how to use the Tube may be a solid life skill.
More recently, the Cable Car over the Thames might be quite cool. I've heard from a few people it is quite good, although that's weather dependent. So if it's a bad day you won't be able to do it.
The other thing I really want to do is 'Up the O2' but I think she might be too young.
London Zoo - although frankly Colchester is better.
kd
The London Planetarium has closed down now (the one that is in the same building as Madam Tussauds. It is now a 4D Marvel interactive ride thing.
I thin you get free entry to it if you book an online ticket to see the waxworks.
rp2000