this is what you do.
walk past her just before you start your workout, then when she is to your side (ie to look at her you have to look over your shoulder 90 degrees, pause. the point is to make it look like youre incidentally asking her, NOT going up to her front on as if youre hitting on her) and say:
"hey (her name), before i start this workout (look at your watch. itll reassure her that you arent going ot hang around forever), do me a huge favour and suggest a nice champagne that women like. one of my friends sisters is having a housewarming, and hes asked me to go along to it to pad out the number of guests so that its a good party, because shes really sweet and would be really hurt if her housewarming party was a flop - but the thing is, i dont want to turn up empty handed and thought that along with a gift for the house, id take a nice wine - what do you think?"
the above story is obviously bullshit, but it demonstrates that 1) you are a good friend 2) your friend is a good guy, which makes you look even better for hanging with good guys 3) you are a good guy for looking after your friend and his sister, 4) are socially intelligent/thoughtful and 5) are not a cheapskate, because youre buying a gift AND wine.
the point of this story is to reopen dialogue with this woman and to give you somethign to talk about - and move on to a quick fluff talk about your/her training, your mom coming to visit you bal bla bla...whatever, talk like you did the other time but as you talk, DO NOT compliment her about ANYTHING and especially not on her looks until right at the end of the convo say somethign like "you know i really regret not talking to you much at the gym, but i focus on my workout really hard when im here...but ive really enjoyed this conversation, how can we continue it?"
she will 90% reply with "well, i could give you my number..." then exchange numbers. if she says nothing or says "i dont know" say "do you have a cell phone or email?" and she will without a doubt give you one of them.
asking her to go on a coffee "date" is pretty much asking her to date you, so its quite loaded...meaning she might say to herself "i dont want to express this much interest to this guy, but i do like him a bit" and say no - even though she wants to hang with you. but if you do what i said then call her and say "hey (her name) my coffee date just cancelled on me - why dont you throw something on and come meet me at xxxxx place. im totally in the mood for a coffee and i would love your company for a half hour or so" then by making each step of the process so small, so easy, theres a great chance youll wind up sipping coffee with your hottie quite quickly
and there you go