I have a 13 yo. I think central London is really very safe - it is incredibly hard to get lost on the tube and there are loads of people to help. That area is crawling with police and apart from the pickpocket risk I think it is probably safer for kids than most other places. Buses are very easy now they announce the stop names. But I would say:
-she has someone else as a backup emergency contact in case of difficulty. A grandparent or friend who works at home perhaps? And she texts them a couple of times in the day to say all fine.
-she certainly doesn't take a blackberry or camera with her. Carrying those around make her far more likely to be the victim of a crime, whether in central London or not. Get her a £10 phone.
-go through likely bus and tube routes and show her how to use the maps. Talk through different ways of getting home.
-they have to stick together, they have some emergency numbers on a bit of paper, and they don't carry valuables. A small amount of cash, oyster cards and cheap mobiles, disposable camera if they want pix - that's it. I would say that applies anywhere in London to kids of this age - they are very vulnerable to casual mugging.
I think it is great they want to explore the city, and they will gain so much confidence.
St James Park is lovely, there is a nice little park next to the Palace of Westminster, and they could walk across Westminster Bridge and then along the river to the South Bank - again very safe, lots of people to help, anyone is able to go into National Theatre, RFH, use the loos and have a sit down. TBH, I'd suggest they start with this area, there is loads on at the moment and it is more relaxed than Trafalgar Square. But even there, they can go into the National Gallery, use the loo, sit down, ask for help in an emergency, have a drink. Possibly even look at a painting.
If the real issue is that you won't be in contact with them and will be worrying all day - which I really understand - could you ask them to do it on a Saturday? The crowds are much the same at this time of year, and there are often things going on at weekends in parks and public spaces in town.