I'm a little confused. I'm assuming you're coming in via I-75. Once inside Florida, there's no good place to stop until Lake City which is about 30-45 minutes from the border. Just north of the Florida-Georgia border is Valdosta.
However, the Florida-Georgia border is well beyond the halfway point from Central Indiana. I live just south of Cincinnati, and to me, the halfway point is somewhere between Atlanta and Macon, GA. That's still a few hours from the border.
That said, you can probably make Lake City if you leave early enough. Our last trip to WDW, we left about 3:30 AM, and got to Lake City about 5 PM.