If I read the calendar correctly, plan your arrival at your rest stop in the Keys on Monday 20160104.
Tentatively, I'm planning to catch the Amtrak Auto Train from Lorton, VA (south of DC) to Sanford, FL (NE of Orlando) on Saturday the second. Two trains run 365 days a year between the two locations, departing each at 1600, arriving the next morning at 0930. One can be on the road by 1100.
With reduced daylight that time of year (in the northern hemisphere) we would choose an overnight stop at an intermediate location Sunday evening. We'd then head down to a place somewhere in the Keys the next morning.