Bit of a silly question, but when she gets up early and raids the kitchen, is it because she is hungry? or bored? Is she actually EATING the food? Or just making a mess? The difference may be a clue in how to help deal with it IYSWIM.
DS2 is 5yo and has always been an early riser (4-5am, but is now leaning towards 5-6am, except occasionally earlier). We have a child safety gate with an alarm on his door, so he cannot leave the room, plus we listen with a child monitor in case he climbs. When he wakes up, he is very hungry, and as soon as he's been cleaned up (wets bed at night), then we go straight downstairs and give him something to start on - usually a banana or an apple. If we don't give him something to eat, he just has a meltdown or gets into everything. With food, he'll focus on that.
If it's hunger, can you put together a snack box that is just for her? For example, a lunch box with some food she is allowed to eat - much like a starter breakfast? Piece of fruit, yoghurt raisins, cereal bar... something nibbly that takes a few minutes to actually eat.
If it's boredom, perhaps a craft or art box that she can use until everyone else is up with safe stuff in it for her to play with?
As far as consequences of behaviour, we've found DS2 only responds well to positive reinforcement and any consequences must be immediate. Anything else and he simply won't connect the consequence to the behaviour.
I well remember the exhaustion - DS2 was regularly waking up 3-4 times per night and often up at 3-4 am for the day up until about the time he turned 5yo, then it has eased off a little. At the same time, DS3 was an infant and waking regularly for feeding - at opposite times that DS2 was waking up of course! And DH worked nights, so I literally survived on naps here and there and that's it. I was so exhausted at one point I was hallucinating. I finally broke down and told DH (he hadn't realised it, as I was trying so hard not to let on how exhausted I was, as he was exhausted too - working nights and only getting max of about 5 hrs of sleep during the day) how badly I was struggling. We worked out some time during the week in the evening for me to sleep about four hours three times per week, to try and get more sleep for me while he dealt with the boys. It made all the difference in the world!! If you can carve out at least 3 hours extra somewhere a couple times a week of sleep, it will help. (3 hours is IIRC one complete sleep cycle) If your DH works days, then try to let him take over from after one child's bedtime, and you sleep for 3 hours that evening.
If there is any way that you can take off a day or two of work (or your DH can), just to allow you to get 6-7 hours of uninterrupted sleep, I promise you, everything will look better as well. That lack of sleep makes everything so much harder!!