If she is capable of sitting quietly for a while and likes the theatre you could take her to see The Lion King at the Lyceum Theatre.
If she likes museums then the Natural History Museum and Science Museum are great for kids - the Science Museum especially.
If you're visiting in December you can take her to Winter Wonderland in Hyde Park which has free entry (you need to pay for rides though). It consists of a German Christmas market, a small funfair and an ice skating rink.
http://www.hydeparkwinterwonderland.com/
As well as Winter Wonderland's ice skating rink there are many other seasonal rinks around the capital. Pre-booking is essential for all of them though as they are very popular! Two locations I can name off the top of my head are the Natural History Museum and the Tower of London.
And, of course, there is London Zoo. Entry is quite expensive but for any animal-mad child it's a great day out.
