So where do they belong? If you say "mental hospital", what's to stop them from simply leaving? If the answer is anything other than "nothing" how is said mental hospital any different from a prison?
I guess the question is, if not prison, what do you do with the people that prove time and again to be incompatible with society?
Prisons are not places of rehabilitation in the US, and a person who is suffering from a severe mental illness is likely to suffer profound abuse in prison.
Anywhere you hold someone against their will eventually becomes fraught, no matter what the stated purpose of the facility is.
At any rate, it's not necessary to pick a specific answer from the massive pile of alternative approaches tried, theorized, and developed in modern history to say that this one thing that is done in the US right now is clearly not the right answer.
Yeah there aren't many people like that and the people who are like that don't belong in a prison.
Anyways, one of the best ways to make someone "a continued drain on society due to continued criminal activity" is to put them in prison, give them a career-killing record that will follow them forever, and put them in contact with other criminals on a daily basis.
Prison is a self-reinforcing system that is itself the drain on society.