Having AC in the lanai is just not possible

I’ve been trying to convert my lanai into a work out room for quite some time. The issue is that I don’t know what to do about AC. The lanai is screened with no windows. The natural airflow and a ceiling fan is what is right now used for cooling. For summer, the air is hot and humid. I drip sweat the entire work out. It gets into my eyes and I slip all over the place. I basically need to get up at 6am to work out and I still am overheated. A cooling system is the solution. The issue is that I don’t know what AC device would work for my situation. Due to the screens, all the AC would fly out the lanai. It would be a waste of energy and very expensive to run. What I would need to do is remove all the screens and change to windows. This way the space is airtight and the AC would not leak out. But then what do I do? Every wall but one would be a window. The other wall is the one that connects to my house. There is a doorway, light fixtures and very small space in between. I don’t believe I can realistically install a ductless mini split air conditioner. I need to mount the indoor air handler on an exterior wall. That is not possible. I can’t exactly connect ductwork to the lanai either. Portable AC would have to sit on the floor and take up space. So what options do I have left?
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