Your question is a little too specific for my knowledge, and the 1860s is a transition period in the United States in regards to mental health. First of all, what do you mean by mentally ill? The 21st century definition is broad and runs the gamut from very treatable conditions (think generalized anxiety disorder or bipolar II) to the profoundly disabling (many types of schizophrenia or schizoaffective disorder). Such a nosology didn't exist in the mid 19th century, and patients would likely be labeled "insane," which would run the gamut from from what we'd today call neurosyphilis, schizophrenia, and dementia (up to a third of patients is asylums at the end of the 19th century actually had GPI, the final stage of neurosyphilis).

So very generally, the "moral treatment" movement started at the end of the 18th century in France and spread throughout Europe (insane people were often chained together in jails) -- arguing for better care for these people, though still often in jails. This led directly to the asylum (meaning refuge) movement, which generally increased the quality of life for the affected -- but were private institutions. Life for the poor, naturally, was still quite poor. Over the middle of the 19th century -- the period you reference, there was a growing movement in the United States to develop public asylums (think Dorothea Dix, one of the biggest reformers in the United States). I do not know anything specifically about Chicago. If you have institutional access, here (https://www.ncbi.nlm.nih.gov/pubmed/3060610) is a good article about the development of the asylum movement.

Sorry that I don't know more specifics and good luck with your question!