Drywall on a lanai is not trivial, we had work done on ours to enclose the glass sliders, they used a product called 'yellow board'. Our neighbors have been having some trouble getting similar work done on a wall on their extended lanai/birdcage, I think they used T&D. Not sure what is going on here, but be careful that you get work done according to the current building code.
We used Kilcrease construction for our birdcage, and the enclosure of the glass sliders was added on to that project. I know they subbed this out, but I do not recall who they subbed to. It might be worth a call,
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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