I'm on the same boat as you...only that I get to see this chick only once a week if Im lucky...
And when I do, shes either leaving or talking to friends
Bad news is that besides hospitals and funerals, the gym is the worse place to hit on a chick,without looking out of place or indiscreet... No matter how secure,smart you look she will perceive you as the "dude " that only goes to the gym to interrupt girls while theyre working out"..
You have to create a situation that'll make you look like youre not hitting on her, b polite and from time to time a bit funny ...dont mean to be a downer but chances to pull it off are low (specially if you have feelings for her).
Good luck though !
WOOOSSSHHHHHHHHHHHHHHH