I've been researching the Exeter School for a good many years. I can't help you find patient records; it's my understanding that much of what once belonged to the institution was destroyed, and the rest disseminated throughout the State - some of it to private collections - after the institution was abandoned. However, I do come across quite a few historical pieces in my work - censuses, antiquarian journals, photographs, newspaper articles, & etc.
I invite you, and anybody with a similar inquiry, to message me with your questions. If you are seeking a specific individual, I would need to know that individual's name and a discharge date/range. In this regard, I cannot promise answers, but I would be glad to keep an eye out for anything that may be of interest.
You may reach me here, on Ancestry.com, or at firstname.lastname@example.org
for a quicker reply.