We had a Mitsubishi AC/heat pump installed in our lanai by M&S AC. The interior unit was placed over the door and the condenser is outside on the patio. It works very well and to us worth the around $3000 price. Room models need to vented to the outside and they didn't work with our lanai windows. You can get an idea of the unit at:
Ductless Air Conditioning | Mitsubishi Electric Product Showcase
You are welcome to look at our unit. Gretchen