funny but stupid
it is simple. When she leaves the gym, go up to the desk clerk and hand them an ear ring or something and tell them that you think it belongs to the lady who just left. They will show you her picture and say is that her? You now have her name. Now that you have her name, next time you are there ask the clerk if she is married, cause you would like to met her but you do not want to bother anyone who is married. They'll tell you.
When you next see her at the gym, you say hey are you "name" and she says yes, then tell her your name and say "I'm sorry I remember your name but not how I know you, can you help me out?" You will then engage in a quick rundown of her school, job, etc. history. And yours: successful, single, etc.(Make it good: "Did you work at Goldman Sachs a few years ago?" etc.) At the end just say, "Sorry, I just can't figure out how I know you but it was nice to meet you again or for the first time, take care". That's it.
If she is interested she will come up to you next time and initiate a conversation. If not, not. This is how the game is played by winners.