The Wellcome Collection is a free museum and library, mostly dealing with medecine and social history. The exhibitions are unusual and engaging, and the permanent collection includes bizarre items such as medieval masks and human remains.
I love swimming and I love being outside. London is great if your into both these things. I swim at London Fields Lido which is heated all year round. Its best in the winter when the steam rises from the water and if your lucky its snowing a little on your head. If your brave you can also go for the unheated option of the Serpentine or Hampstead Ponds.