Travelling to London in winter

I am tentatively planning a one-week vacation in London over Christmas. Can someone please advice on good places to stay (preferably close to center), and what are the "must-see" (perhaps something that would not be mentioned in a tour-guide book). All the pointers will be greatly appreciated!

Also, is anyone aware of any show-jumping facilities with good trainers nearby (well, I know its too much to ask...)

:) Natasha