Do you like walking? I might be tempted to head for Westminster in the morning and then work my way by bus/tube and foot to Paddington for the express back to Heathrow.

Although I love the area around Bermondsey it's definitely not worth paying to get into teh Design Museum and there are just as many interesting things going on elsewhere.

If you do fancy having a look around there I might head for Tower Hill or Bermondsey on the tube and then take a river tour from Tower Bridge East or West, you see loads that way and learn a bit about what you're seeing too.