Honestly, my favorite thing to do in London is ride around on the underground and pop out in different places to get a feel for the entire city (zone 1 at least). I suspect you sympathize. But if you have a day, boats leave from the downtown docks heading to Hampton Court. Really soothing trip along the Thames, and you can tour the castle at the end. Take a train back to save time. 4-5 hours. And if you really have time for a day trip, go to Oxford. It's the college town. If the weather is good it's one of the most peaceful places on earth. Bus trip is ~90 minutes one way, I recall, and cheap. If you have limited time, simply wander around Hyde Park at night.
That's a shame. I'd maybe skip the museums, if you only really have one day. Just explore, eat at a pub, drink ale, look at the famous buildings on the Thames and wander the famous gardens/parks. And! Put the Knowledge to the test if you have some funds. I bet any cabbie you meet would enjoy talking about it.