I often build a 12-14 hour layover at MIA into my trips back from Brazil. Arrive in at 5am or so, in the AC by 6am. I usually find a plush chair in a quiet corner and sleep 2-3 hours till rush hour is over. Then a quick refreshing shower and I'm ready to spend the day in Miami. I rent a car and either meet my friend for lunch if he's in town or head to Aventura Mall an mooch around the shops picking up the stuff that's cheap in the US.

It doesn't really answer your question but illustrates you can put the day to good use. If you want to crash just get a cheap airport area hotel on Priceline. If you'd prefer to spend the day in the airport then IIRC the Airport Hotel has leisure facilities on the 8 or 9th floor which you can pay to use.