Its a fair way from Orlando and is about 400 miles to Key West. We have done it a few times and it probably takes about 8 hours with a short break. Big problem is when you get past Miami the roads narrow and your average speed drops.
If you are setting off from Orlando virtually as soon as you get there then the best bet could be to stay around Ft Lauderdale for the night and break up the journey.
Weather in January can be variable but its usually warmer in the Keys than in Orlando.
We usually spend time in Florida at that time of the year and its good. Not too busy and the chance of decent weather.

To be honest I wouldn't recommend it in a day.
And of course it also dependes on where on the Keys you want to stay. We went from Orlando to Ft Laud (were we stayed overnight) and then onto Key West the following morning. We chose to stay on Key West first and work our way back along the Keys and if I'm honest, the journey from Ft Laud to Key West made it a long day.

weather this January in Orlando has been a bit chilly-but to be honest we haven't had much rain-so much rather here than in UK-and today has been t shirt weather.