There are many stories about Johnny Carson at the Mai-Kai, and especially stories about Johnny and the Mystery Girls, but here's one I just heard from watching the new American Master's on PBS about him. It may be his first visit all the way back in 1962.
In 1962, right after being hired by NBC to take over the Tonight Show (from Jack Paar), Johnny, Ed McMahon, and his producer went to Ft. Lauderdale for two weeks to work on the show. Apparently one night they went to "a Polynesian restaurant" and had too many Mai Tais after which they got up on stage, put on grass skirts and got hula lessons from the performers.
The restaurant was not named, but especially with the stage mentioned, I was guessing it was the Mai-Kai. A little digging found a confirmation of it being the Mai-Kai in the book "Ft. Lauderdale: Playground of the Stars" by Jack Drury.
http://bit.ly/KqoA7f
Catch the whole American Master's episode if you can. The Mai-Kai Mai Tai story is about 27 minutes into the show.