When you say that she "will not" what you really mean is that you won't. If you sign her in at the gym and walk away - she will have no choice. She may get upset the first few times and might even cry hysterically the whole time. You need to remember, your job is to develop a happy adult and that means that you will sometimes have an unhappy child. Focus on the ultimate result rather than the immediate situation. It is not healthy for your daughter to be with you 24/7 she needs to develop healthy social interaction and you need to provide those opportunities for her, even if she doesn't "want" to at the moment. When you drop her off, don't linger and coddle her or show that you are upset about leaving her. Don't apologize! Your daughter will take a cue from your emotions and reactions so she has learned that if she throws a fit, you won't leave her. When you drop her off, simply say, "It's time for you to play with some friends... I will be back in a little while to pick you up." Give her a hug and hand her off to a staff member. Then, don't look back!!! Go work out and then come back and keep your tone upbeat as you ask her about what she did while you were gone. One technique I have found helpful is to be as casual as you can, but be firm.
Pick a certain day of the week and time of day so she can learn the routine and do it the same way every time. Once she realizes that you are consistent in the drop off and pick up process, she will realize that you always come back to get her and will begin to explore the fun activities rather than manipulating you. You need the time alone and I know it's hard but remind yourself you are doing it for her wellbeing as well.