I usually get the coach to Stanstead. You can National Express or Terravision, and they run all night. There will be ample room for your luggage and other stuff, and the kids will be able to sleep. At that time in the morning the journey is usually a breeze, and is in my opinion much easier that mucking about with trains. You get dropped right outside the terminal too.
Not sure where you are coming from, but you can just get a cab to Victoria Coach Station and jump right on. It's probably a good idea to get the tickets in advance if you can.
Vicky