There are way too many things to see in London to list, but very few of the tourist sites I would personally suggest would be 'Christmassy' themed, which it sounds like you are looking for. In the nicest possible way, I would not go to London to see Christmas tat when there are so many other things available, but I don't have kids to be fair.
With kids in mind, Hamleys, Regent St lights generally, Oxford St shopping and lights, Covent Garden buskers/jugglers etc and all the usual west end tourist traps will be nice enough if you don't get into town much.
If you can use the train, use it.
Driving in London is utterly pointless if you can get there by train particularly when you don't know where you are going. If you drive, it will likely ruin your weekend.
If you want to see an evening show, then pick somewhere to stay relatively near the theatre if you don't want to travel great distances at night without knowing where you are, but be aware they do know how to charge for hotels in that area!