Having A/C in the lanai is just not possible

I’ve been trying to convert our lanai into a work out room for quite some time, then the issue is that I don’t know what to do about AC.

The lanai is screened with no windows. The natural airflow & a ceiling fan is what is right now used for cooling. For summer, the air is hot & humid, however i drip sweat the entire work out. It gets into our esure & I slip all over the place. I basically need to get up at 6am to work out & I still am overheated! A cooling idea is the solution. The issue is that I don’t know what A/C device would work for our situation, however due to the screens, all the A/C would fly out the lanai. It would be a waste of energy & very extravagant to run, what I would need to do is detach all the screens & change to windows. This way the space is airtight & the A/C would not leak out… But then what do I do? Every wall however a single would be a window. The other wall is the a single that links to our house. There is a doorway, light fixtures & very small space in between. I don’t guess I can realistically install a ductless mini split cooling system. I need to mount the indoor air handler on an exterior wall. That is not possible. I can’t exactly link air duct to the lanai either, and portable A/C would have to stand on the floor & take up space, then so what possibilities do I have left?

Heating technician