My house is 100 years old with no exhaust fan in the small 7'x9' first floor bathroom. This is a problem because it's the only bathroom with a shower. The bathroom has a window, but it opens on to an unheated Florida room. Is venting up two floors & then through the finished attic too far of a distance for an exhaust fan? Another option would be to go through the bathroom ceiling, have the vent duct take a 90 degree turn & just vent it out to the Florida room. This room is 9'x 20'. Is this a bad idea? The only alternative I can think of is running a duct along the ceiling of the Florida room with a vent to the outside, but that wouldn't be very attractive.

Thank you for advice & help.