Three levels of lanai enclosure
Screened keeps insects out and rain mostly off. Cheapest, and still an outdoor room.
Glazed but unconditioned is weather-tight, usable most of the year, and genuinely uncomfortable in August. It does not add to the conditioned square footage of the home.
Conditioned makes it part of the house. That means insulation, HVAC, and the same envelope and code requirements as any other room. It also counts toward square footage. So it is an addition, and it takes an addition’s permit.
What the existing structure can carry
Builders often framed an original lanai roof only to shelter an open slab. It rarely carries enclosing walls, or the loads the Florida Building Code assigns an enclosed room. Whether it can is a structural question, and it gets answered before design. Sometimes the roof joins the project whether you wanted it to or not.
The slab
Crews usually pour lanai slabs to fall away from the house for drainage. Enclosing means handling that slope, and the transition at the door threshold. Gloss over it and water problems start there.