Not really understanding what her waffling is about. She is allergic to dogs to the point her health is suffering including coming in contact with her child who has been around a dog. She needs to move her child to a home without pets and deal with her own pet. Hanging on till she finds something else, hoping it will go away, not sure what she is waiting for.

Especially considering she is part time, a teacher that doesn't need care all summer so has plenty of time to find something for Sept I would be inclined to terminate her as of the end of the school year and advertise to fill the spot. I would use the reasoning that for her health you realize that she doesn't have a choice but to move to a pet free home and that you wish her well but will be available to her till end of school year.

That allows you to advertise for a full time child so you come out ahead but really what choice is there if what she is saying about her medical condition is true. Right now she is in denial that she needs to get rid of her own dog. But even if she doesn't leave care now she should over the summer.