Short term rentals can be very difficult to find on Miami Beach these days. The city of Miami Beach collects a hotel/motel tax of 14% on any lease of 180 days or less. The person who receives the rental income is responsible for paying the tax. Many HOA's have extended the mininum lease period to avoid problems associated with collecting the tax.
I had a meeting today with a company called Design Suites who manages over 100 units at Castle Beach on Collins Avenue (The old Playboy Club). They have studio, one and two bedroom units, fully furnished, and prices start at $135 per night during high season. They do not add on a cleaning fee at the end of your stay. This is a great value compared to Miami Beach nightly hotel charges. All units they manage have been refurbished, and include fully equiped kitchenettes, safe box, Ipod docking station, iron, and hairdryer.
The building has a turkish bath, fitness center, pool, laundry, and dry cleaners.
Let me know if you need additional information on this property.