Hi Sherry, and welcome back to the Disney Parks Moms Panel!
Looking at the Park Hours calendar, I can see your dilemma. It does seem unusual to find a day without Extra Magic Hours available at any of the Theme Parks, but it appears that January 2 may be one of those odd days.
I know that Park Hours are always subject to change, so you may try to keep an eye on the calendar to see if any Extra Magic Hours have been added. If Extra Magic Hours are not added on that day, you can still have a great time in the Parks! In fact, many Guests avoid Parks with Extra Magic Hours due to increased crowds. Just verify the operating hours at the Park you want to visit with your Resort concierge and be there early to catch "rope drop," or the moment the Theme Park opens. You can usually hop on many attractions with much shorter waits that way. Combined with late-morning or early-afternoon Fast Pass+ attractions, you can have a magical time even without Extra Magic Hours.
If you were looking to spend a late night in a Theme Park, you might try an evening at Disney Springs. With all the new food and drink options there, your daughter can have a blast on that big birthday!
I hope this helps with your planning, Sherry. Please stop by and see us again if you have any more questions.
Keep Disney Dreaming,
Kathy K.