Actually, I believe that (depending on which state you are in) they are required by law to keep all records for a certain amount of time after closing. Usually years.
The short answer is "it's impossible".
Obtaining psychiatric records is almost impossible for an individual, and as for a third party - they won't even release them to another physician - only another treating psychiatrist. Psychiatric treatment records fall into a special legal category.
Then there is the matter of destruction of records when an institution closes. Unfortunately the patient records are usually destroyed unless the institution merged with another.
If they do exist your only real option is to have them requested by another psychiatrist who will agree to provide you with them - very unlikely.