Hi & Welcome. How old is your daughter?
Many schools will not give an overall diagnosis in this case because it really isn't within their qualifications to do so leaving developmental issues of unspecified nature. What I would recommend doing is calling the developmental pediatrician's office and ask to be placed on their cancellation list. Tell them that you are flexible and then be prepared to move on short notice. Cancellations don't come along often but if you do as I've recommended you will stand a better chance at getting an earlier appointment.
Usually there is an underlying neurological condition for the cluster of behaviors that you are seeing. If there was maternal drug use during pregnancy that could be involved. We do see a lot of undiagnosed high functioning Autistic Spectrum Disorders and Early Onset Bipolar coming through here just to give you an idea. Both can be very difficult to diagnose at early ages so keep your mind open, your eyes open, and do your research thoroughly.
You mentioned her perservating over something she gets stuck in her head. Other than this are you seeing obsessive or compulsive behaviors of any kind? Is she insisting on wearing only a certain outfit or eating only certain clothes?
You will help yourself on the homefront if you pick up a copy of the book The Explosive Child by Ross Greene. The book is written for older children but at the top of this board you can see how we use the principles adapted to younger children.
I don't mean to discourage you but it's probably going to be extremely difficult to find a day care setting that she's going to fit into at this point. Follow through with the school board but visit our special education board first to get some help on how to advocate for her. Many schools are reluctant to provide a 1 on 1 aide for a private preschool setting but it can be done. If you could find someone to come into your home for daycare that would likely be the most suitable setting until you have a grasp on what is going on.