L.,
I doubt this has as much to do with teething as it does with her new found developmental skill of climbing out of the bed. Once they can get out of bed sleep can become more challenging. As a Child Development Specialist and mom I would suggest:
*give her lots of chances to climb during the day to practice this motor skill and also tire her out in a good way
*put her mattress on the floor or give her a bed so she does not fall while climbing out
*put her to bed and put a gate on the door to her bedroom so she can't get out of her room which may mean two gates to cover the door since she can climb well
*put her to bed awake and DON'T wait with her until she falls asleep,part of learning to sleep is falling asleep alone when tired
*when she cries call to her every five minutes without her seeing you and say, "Mommy is here and it is time to sleep". Don't say anything else and keep your voice calm. Watch the clock because it can seem like a long five minutes. I did this with my son and the first night I went in 5 times and the second night once and then we were done.
*If she sleeps for the sitter that means it is behavioral. I often hear parents say kids sleep for sitters and not for them. Keep the rules the same and don't feel guilty!
*If she gets up at 4AM I would repeat the pattern of the night but if it is 6AM I would let her get up and then tire her out for a nap and repeat the night routine for nap if you have to. If you know your child is tired that is all the information you need to know you are doing the right thing. It is a battle of wills and this point and she needs to know who is winning this one.
Good luck. Let me know what works. I am very interested in sleep behavior. A.