Abby,
How frustrating!
A thought I had...I've heard that seaweed can help to regulate a thyroid that goes up and down like you have described, and seaweed doesn't have side effects like medication. Of course, don't have her try this on my suggestion alone, talk to a doctor first, maybe even do some research. I don't know enough about thyroid problems to know personally if this is helpful, just remembering what was said by other people with thyroid problems. I imagine that eating more iodine in some cases could be a bad thing to do when it comes to the thyroid.
I wish your daughter the best.