Thorpe Park, Legoland and Chessington are all fairly accessible on public transport from London - the journey time from central London for each of them is (very roughly) about 30 minutes on the train. You can easily get there and back in a day - no need to stay nearby overnight unless you want to. You can walk to Chessington from the train station, for the others you need to get a bus which takes another 15 minutes or so. Do you know where abouts in London you're going to be living? The actual journey time will obviously vary massively depending on how far away you are.
Alton Towers is possible to get to on train and bus, but it takes forever from London and can't really be done in a day - for that one I'd recommend booking a hotel or B&B nearby and staying there for a night or two.
The only major restriction on the London Eye is that you can't use your pass on it in August. I think the same applies to the other central London attractions (Madame Tussauds, the London Dungeon, the Aquarium) - frankly I think all of these (except the London Eye) are rubbish anyway, there are much better things to do as a tourist in central London.
The actual train stations you want to get to are:
Staines for Thorpe Park (then 10-15 minutes on a bus)
Windsor and Eton Riverside for Legoland (then 10-15 minutes on a bus)
Chessington South for Chessington (10 minutes walk)
Stoke on Trent or Uttoxeter for Alton Towers (then 30 minutes on a bus from Uttoxeter, or about an hour from Stoke on Trent)
For train times and prices, go to http://www.nationalrail.co.uk/ and tell it where you're planning to go from and to.
One thing to remember before buying a Merlin pass is that you can pretty easily get "2 for the price of 1" vouchers for all of the Merlin parks, so if you're going with someone you'll need to go about 7 times for it to be worth the price.