I think what you might need to do is break down your research into two angles--the prison side and the mental health facility side. I have no knowledge of the prison system, or how it is applied to the hospitals. But I did work extensively in residential mental health during college (psych major). So I can give you the therapeutic perspective, and maybe someone else can chime in with how the prison aspect changes things.
Residential mental health facilities live and die by schedules. The idea is to maintain as much routine as is humanly possible, and to keep the residents productively engaged. So there is always a big group schedule prominently posted with start and end times for meals, group therapy, arts and crafts, medication distribution (usually 3x/day, not all residents get meds at every distribution), even personal hygiene times. Then each individual client also gets a personal schedule (individual therapy, phone calls, doctor appointments, personal goals). Naturally, most residents aren't too good at keeping up with times and following schedules on their own, so a large part of the average mental health tech's day is taken up with corralling people to the next thing on the agenda. But routines help them feel safe and help keep them calm, so they're absolutely vital.
In regular residential mental health, the overarching therapeutic goal for each resident is stabilization, with the goal of returning to the community as soon as possible. Obviously this is different for a prison hospital, and I don't really know what the overarching therapeutic goal would be, since they're not returning to the community anytime soon.
Rewards rather than punishments are the preferred method of behavior control, at least in regular residential facilities. Pretty much anything you can imagine could be a reward--TV time, extra phone calls, participation in an evening group activity. Different levels of rewards call for different amounts of good behavior--a resident might get a half hour of TV time for following the day's schedule without complaint, but might need to actively participate in group therapy all week to join in a Saturday afternoon game. It's all very individualized to the resident.
In mental health as a whole, a good therapeutic environment is stressed--pretty colors, green spaces, plenty of exercise...but again, I know this is the polar opposite of thinking in the prison system, so I don't know how it translates.
Anything can be a weapon, so staff are always cognizant of that possibility. You haven't lived till you've been stabbed repeatedly in the arms and legs by a 9 year old with a pencil. Accordingly, residents tend to have everything potentially sharp or dangerous taken away when they come in, and very gradually earn the privilege of using pencils, shoelaces, etc. Just because someone earned the privilege doesn't make it "safe," however, and doesn't mean they won't snap--had a guy in a transitional facility, just about to be released, decide to stop taking his meds, steal a knife from the kitchen, cut himself up, and then go after a few of us. It was back to the hospital for him. The constant struggle for staff is finding the balance between building therapeutic relationships and keeping a guard up at all times.
Every resident goes through an intake process--you get their file, search them for contraband, give an orientation tour, assign a bed, give them a packet of information, and help them get settled. It usually happens as soon as they come in, though it may be abbreviated if they come in during the night, and finished the next morning. They see a psychiatrist within 24 hours, at least in my experience, and get assigned to a therapist in about the same time period.
Again, all of this is based on residential mental health that is not associated with the prison system. But hopefully it at least gives you a starting point. I'm happy to answer questions as well.
