Not sure about just the mentally ill, but the Alexian Brothers are specifically devoted to the care of the sick. They provide services to those with psychiatric disorders.
Perhaps the Order of Saint John of God? Care for the mentally ill is a major part of their mission - St John of God (1495-1550) himself spent some time as a patient in a psychiatric hospital in Granada, Spain. He prayed to God that if he was released from the hospital he would open a hospital that would offer dignity and respect to people. The website of the order worldwide is: oh-fbf.it/Objects/Home1.asp
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