Sorry I couldn't come out to this sooner. I spent some time this morning searching the web and making a phone calls.
Here's the deal:
The archived records are at the hospital (after a few moves). The records department will not release any records for genealogical purposes. You could get around that, though, by having your doctor request the discharge papers for your relative. Of course, there's no guarantee the records you want still exist.
The records moved from the City Sanitarium to the Missouri Institute of Mental Health Library, then that Library closed for a while. At the time the MIMH Library closed, they sent the records to the St. Louis Psychiatric Rehabilitation Center (the new name for the City Sanitarium).
So, unless that Public Library lead bears fruit, you'll have to see if your doctor will request the records for medical purposes.