Thank you for your question and welcome to Just Answer.
Your daughter has an infection of the tissue flap that is partially covering the erupting tooth. Food has percolated under the flap, too far removed for her to clean out with her tooth brush. You are doing all the right things but she needs to be placed on some antibiotics. This situation will take care of itself as the tooth continnues to erupt and the flap dissolves away. But for now, she needs the assistance of the antibiotics t reduce the swelling and eliminate the infection. It's your call whether to go the toe ER just to get a prescription as that is all they will do and it's an expensive way to get one. Tomorrow morning her dentist can call one into a local pharmacy for you and she can start taking it at that time.
I hope this answers your question. If you would like to discuss this further or have any additional questions, please reply to my answer and I will get right back to you.
Sincerely, XXXXX XXXXX DDS
You are most welcome. I wish you well.
SIncerely, XXXXX XXXXX DDS