There aren't boarding schools specifically designed for people with mental illness in the way that there are boarding schools for other purposes (like academics or special needs). The treatment of mental illness requires a wide range of interventions, and these are usually provided in specialized settings like hospitals, residential treatment facilities, or community-based programs. These facilities differ significantly from traditional boarding schools.
While some residential treatment facilities for mental health might have a boarding school-like environment, the focus is entirely on therapeutic care, not academics in the traditional sense. The level of supervision and the therapeutic goals are very different.