How do I get my brother the help he needs?

I agree with Touchmatters. Talk to the hospital social worker, at least he or she can point you in the right direction.

Nursing home.
Response below very helpful. Talk to hospital social worker.
Is he on Medicaid ?

Gena / Touch Matters

Speaking to a social worker at the hospital or rehab center, express that he will be an unsafe discharge. As for insurance and depending on the state where he lives, the social worker can assist on him getting disability insurance which is similar to Medicaid. Especially if it is permanent. He can also apply for Social Security disability but it is a longer process of 6 months last I looked into it.

I agree with others that he needs to contact social services for his county. If he's employed, or was employed, he may need to apply for disability as well.

Your questions need not to come to a Forum of strangers but to do directly now to social services. They should have been reaching out for you from day one in hospital both to get your brother on some insurance and to also get him what care for the indigent can be found. Go now to social services whether hospital or rehab and begin to tell them the truth of his life. You need to be ready to submit his bank accounts and etc. This is something that is going to have to go through a lot of social services help. I wish you the best of luck.

You or your brother are going to have to let the rehab center know upfront that he will be an "unsafe discharge" until he is strong enough to take care of himself as he has no family that is willing or able to take him in or care for him.
As long as you both continuously use the words "unsafe discharge" the rehab center will then have to find placement for him in the appropriate facility and Medicaid will have to applied for.