I live maybe 10 miles north of Hollywood, FL, and I've been fishing in that area, and farther offshore, on and off for quite some time.
I find the idea of a submerged submarine in that area a bit hard to believe, and the idea that he was doing anything resembling 20kn nearly impossible. Ive never seen the navigation charts of the area, but my rough knowledge is that the seafloor off the southeast Florida coast is VERY gently sloped. between 3-5nm offshore, I don't think there's many (If any) places where its more than 100-200ft deep.
In water that shallow, I don't see a USN nuke doing more than a few knots, and I seriously doubt anyone else would risk being that close to shore, seeing how there's nothing militarily important to look at, and you're well within US territorial waters. (Lots of NON-military things to look at, though, Particularly if you've got a good magnification factor in your optics...
)
Either the observer has terrible recollection skills, is lying through his teeth, mistook what he saw, or the North Koreans are coming.
We'll probably never know, though; If it was a USN vessel, they'll never admit it, and if it wasn't, they'll also never admit it.