We went the last week of January this year. It was so cold there was ice in the GF fountain. It was one of the coldest spells ever in Florida. Being snowbirds (Dayton,Oh), we found the weather to be pleasant.
We love January!!!. No lines, instant seating in most restaraunts, no crowds, the parks still have all of the shows, no crowds, 5-10 minute wait in Stand by, able to actually see the parades without waiting for hours on the curb, no crowds, no sweating, no heat exhaustion, no crowds.
If you are a swimmer you may be disappointed. Though the resort pools are heated, we didn't swim. Others did, it was funny seeing people swim in the steam. A few times a family of ducks would land in the pool and swim in the steam.
January park hours can be short, but with no waits you'll be able to fit most things in.
Also the on-site hotel rates are at their lowest in January. The only problem we see in the future is convincing the kids teachers that this is an educational experience.
Go and have a great time. Get a chuckle from the locals who are all dressed in wool coats in 60 degree weather, while you go in short sleeves and jeans.