How long can a hospital hold a person they say is a danger to themselves?
They even admit that the person is not suicidal.
Whether or not the person is suicidal isn't the question; the Michigan Mental Health Code has a moderately complex standard, and the finding doesn't have to be much more than that it's likely that the person would cause himself significant physical harm or would be unable to provide himself with food and shelter because of his mental illness.
The time limit is 24 hours. By that point, the person either has to be released, or the hospital has to provide written notice to the court, and to two (2) people designated by the person, that there is involuntary confinement, and that triggers a judicial review process, during which the patient has the right to the assistance of an attorney.