You work where I used to work. It must be the same place as there can't be two facilities in the country with the same problem, right?
It seems to me that the problem with managers in aged care is twofold. The first problem is that most managers are nurses by profession and good nurses do not necessarily make good managers. The second problem is that some of the people we see currently in aged care management weren't even good nurses to start with. The recent history of aged care as a second rate profession means that it has unfortunately attracted second rate nurses, or at the least, nurses who were there for reasons other than wanting to be in aged care eg. set shifts, or flexibility, or even the view that it was "easy" work. (And we all know it's easy to do badly if you are task orientated; drug round, dressing round, scant documentation, go home. I say all of this with apologies to all the other excellent nurses and nurse managers out there whom I work with every day, no matter how they came to the profession.)
Anyway, back to my last place of work. I suffered through seeing a colleague bully staff, residents, and relatives. She assaulted several residents, as well as being known to have thieved drugs. I (along with countless other staff, residents, and relatives) reported the incidents to management who refused to act, even on the assaults. Even written complaints to the CEO went unheeded. To this day I am unsure why this was so in the face of obvious incompetence, malpractice and unprofessional conduct as well as the associated problems you mention of lost productivity, staff resignations, and agency refusing to come in. In the end, I had no choice but to resign, as I felt it was ethically and professionally the right thing to do.
Now bear in mind, that I was a senior member of registered staff, with plenty of resources, a forceful personality (!), and a couple of degrees behind me, and then consider that I got exactly nowhere. I feel desperately sorry for the more vulnerable junior staff and residents.
You have some choices. You can do what I did and leave. Or you can leave and whistle blow. Or you can stay and whistle blow. If you choose to whistle blow you need to be aware that there is legislation which protects you but the harsh reality is, it probably won’t help much. So you have to make an ethical and practical decision about what the right course of action for you is. If you do choose to whistle blow, I would advise you to document everything. You need to record as much detail as possible, not only times and dates, but try to include things like tone of voice, stance, the exact words used etc. Make sure your reports are factual and use correct grammar, spelling and punctuation. Unemotional and well-written reports carry more weight. It is also very important to document the times you have reported the behaviour to management and what their actions were.
Who you go to with your information depends somewhat upon what sort of employees these are. Unprofessional conduct by Registered and Enrolled Nurses can be reported to the Nurses’ Board in your State. This has the added advantage of dumping management in it too as they have an obligation under the legislation to report unprofessional conduct (as above, make sure you document management’s actions too). Taking or limiting their registration solves all the problems in a very effective way.......

Outside of the Nurses’ Board, I think the only options left to you are the CIS and Accreditation people, and quite frankly they are toothless tigers which I wouldn’t be bothered with.
Others may have more ideas.