OP, remember the lanai has a sloped floor, ours is slightly larger than yours, but we have a 2-inch slope over the 13 feet. Depending on exactly what you want to do in this room this slope could interfere. You may also want to consider additional HVAC for this area, the mini-split systems seem to be in the $4000 to $5000 range. No idea on the enclosure costs. Also, make sure you maintain one or more exits, check building codes on this one.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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