This would be a lot easier if you had something in writing that showed you cancelled within the time allowed. As it is, it is your word against theirs, and your guess is as good as anyone's about how that will turn out.
If you can document your claim in any way (cell phone bill showing contact perhaps?) I'd feel better about this, but again, without direct proof you cancelled the account, you have an uphill battle. These companies don't make money by letting people out of their contracts, so they always play 'hardball' with people in my experience. You should document in writing NOW (at the very least) that you cancelled the membership and put in specific information such as the date, time and name of the person you spoke with.
If this doesn't get them to back off, you may have to defend yourself in Court if they sue you. I'd suggest consulting with a local lawyer as soon as possible (even before you send the letter above if at all possible -- which you should send within the next day or two at the latest!)
And for the future, please remember, if it didn't get reduced to writing, it didn't happen. Good luck!