And then there is also Alexandra Palace, but that is in north London. You would have to travel to work by tube, about an hour from Hammersmith by tube.
Just want to comment that I'm happy to hear that those rinks are still there. I did some training at Richmond Ice Rink in the mid 80s. That was such a lovely rink and it made me sad to hear that they took it down (always figured that would happen some day).
I remember Streatham also being a very nice rink.
Enjoy your time there leafytoes!