That should get us close, but I don’t think it is possible to reach zero homelessness. There always will be some vagabonds who will wander around from short temporary job to short temporary job, and occasionally will sleep on the street. Always give them the option of temporary shelter, a place to clean them up, but as long as they don’t do harm to society, don’t force them to have a home.
Yes, vast increase of capacity of prisons is necessary, today it's difficult to fight many types of crime because there's nowhere to put criminals in - vast, gulag-style camps able to house as many people as needed, is necessary. Make sure they don't come back until they are too old to be a problem.
Seriously, society needs some discipline and some fear factor. Most people can be motivated by greed, but not everyone, so some healthy portion of fear is necessary.
I always make an example of Belarus as a country that solved homelessness. It is a crime there, but extremely small number of chronic bums are actually behind bars - rest are too scared to even try bumming so they try as hard as possible to hold on to work, housing and their families.