Welcome to FT rhardber and zechristof. I would agree with zechristof that the tube would be a better travel option than the Heathrow Express - it may take a little longer but it gets you much closer to the bits of London you'll probably want to see.

I'm going to shuffle this thread over to the London forum (which is a subforum to the UK forum) - take a look through some of the threads there and you may find some suggestions for what to do with you time.