You should just fly from london to amsterdam. Catch a bus to luton airport, or a train to stansted (either takes just under an hour), then catch a budget flight for about 40quid to amsterdam, the flight is only an hour and usually leaves around 4pm. check out of your hotel at 11 (normal checkout time), get a bite of lunch and say goodbye to london, cruise to the airport, and you'll be in your hotel room in amsterdam by 8pm, guaranteed. Trust me I've done it and plan to again in august.

Then If you're set on Paris (it's dirty and expensive), catch the eurostar from the dam and you'll be there in just a few hours. None of those trips will take up more than a half day.

As for things to do in Amsterdam, I plan on spending a lot of time in the vondelpark this time since i'll be there in summer. They have free concerts and movie screenings on there all the time, and it's just a great place to relax and smoke on a nice day. The entire city is awesome to hang out, shop, visit bars (and coffeeshops obviously), and eat in. It's one of my favourite cities in the world even without the smoking and red light district.