Many employers would have company policies for their employees about what they can or can't photograph or video in the workplace. If this is at a small locally owned daycare with just one location (as opposed to a larger one with multiple locations), perhaps there in not a company handbook with all the policies spelled out... Check to see if there is, and also check whatever paperwork the parent customers have to sign when they sign up. My guess is that there may be language somewhere that addresses privacy issues. A daycare should be a private institution where someone would have an "expectation of privacy" as opposed to if you're child were walking down a public sidewalk and someone took a video of her...
I don't think that employment law would really address this issue. I suspect it comes down to a potential violation of company policy and your privacy rights. Especially if her motivation in posting the video was to embarrass, mock, or make fun of your daughter. Perhaps different if she thought it was cute, but still inappropriate to do so. Any idea what comment may have been posted with the video?
My guess is the supervisor/owners are checking with their attorney to see what their best move is. They may be looking at various punishments for the other employee, what their obligations are, and whether they need to update their company policies.
You likely have potential legal action you could take against the teacher and/or the daycare. But I hope this is something that could be worked out privately with an apology to you, public apology, and/or fine/suspension/termination of the employee. If you want to consider legal action, your best option is to consult with a local attorney. If you have specific questions, please let me know, and I will try to answer further.