It doesn't work because homelessness is a structural feature of our economy. Until we decide to return to the level and type of wealth redistribution we had before the 1980s, it's going to remain about the same.
"In his first year in office Reagan halved the budget for public housing and Section 8 to about $17.5 billion. And for the next few years he sought to eliminate federal housing assistance to the poor altogether."
From the Wikipedia article entitled "Public housing in the United States", which has a section specifically about the period of time in question:
"Under the Reagan administration, household contribution towards Section 8 rents was increased to 30% of household income and fair market rents were lowered. Public assistance for housing efforts was reduced as part of a package of across the board cuts."
I am not an American and don't know the history of American federal program funding, but I do have access to things that let me search for things. You should try it.
http://www.cato.org/pubs/pas/pa127.html
#Edit. Reading the article further, it seems the Cato institute dismisses/rebukes most of the points regarding federal housing being the cause of homelessness in the 1980's.
From the link:
Like the shift from 40-year mortgage subsidies to 15- year housing certificates, the shift from the certificates to 5-year vouchers enabled Congress to reduce HUD's budget authorizations. Such reductions have created the mistaken impression that federal housing assistance has been cut. In fact, annual outlays increased from $5.7 billion in 1980 to $13.8 billion in 1988. The number of families being served by HUD housing programs rose from 3.1 million in 1980 to 4.2 million in 1988. Moreover, a housing voucher costs the government only about $4,500 a year, whereas a new unit of public housing costs it $9,000 a year.
I guess mistaken impressions are still common on the internet.
1) most homeless people do not have a mental illness. MI is over-represented in the homeless population, but only one fifth of rough sleepers have a mental illness.
If you look at the chronically homeless (people homeless for over a year) you see that a third of them have a severe mental illness
If four fifths of the homeless population or two thirds of the chronic homeless population are not mentally ill it's hard to see how mental health hospitals would have made much difference.
2) that people with a mental illness must have a hospital bed and nothing else will do. The UK also closed many large hospitals - currently only about 8% of the people accessing specialist mental health services need a bed. Most people need treatment in the community. It's safer, more effective, and cheaper.