Can a priesthood holder bless his sister who is mentally sick and is not a member of the church and therefore can not for herself ask to be blessed because of her said condition? Thanks.
Faith can be exercised on behalf of a loved one. A man approached Jesus and asked that his son be healed of the many afflictions he suffered – self-harm, foaming at the mouth and dumb (Mark 9:17-29). Note the son was in no condition to request this special blessing on his own, but needed the faith of his father on his behalf. Jesus respected whatever faith the father had, finding even a little faith sufficient for this miracle, and cast out the possessing spirit.
I need to also point out that depending on the form of mental illness, a person may still independently demonstrate sufficient faith to be healed. Consider the case of many young children who are not denied their Heavenly Father’s ear on account of their physical or mental immaturity.