I dont think the gym is a good place for a young child to hang out. Other people working out aren't going to want to watch out for kids (most gyms wont let kids under 14 or 16 near the equipment). Spending a little time in a childcare area while you work out is one thing, but if you want her to be able to run around, why not take her to an indoor playplace or a museum? Some malls have indoor play centers, too.
I used to live in Chicago and Green Bay (you might have heard about their weather recently, 20 below with windchills at 40 below) and it's too cold to even think about playing outside. I used to take my son to places like Chuck E Cheese, or McDonalds to play in the indoor play area. Sometimes we'd have lunch at McD's, but sometimes we would just buy a dessert (ice cream in winter, we have issues!) then play.
You could try your local community center or YMCA - they will have acitivities (and equipment) geared towards a young child so that she can run around in a safe environment and burn off some energy. Also, as long as it's above freezing, don't be afraid of the outdoors. Bundle her and yourself up in layers and take her to your local park for a bit. Depending on the temp, she might only get 15-20 minutes of playtime but fresh outside air has a way of tiring kids out :)