Can a non-relative Wis. resident take temporary custody of a child whose single parent Illinois resident mother is homeless?

The Illinois mother is in the process of divorce and is currently homeless. Her husband is not the father of the child and has not adopted the child. The mother of the child is living with a man that is not her husband and his mother in a weekly rental motel room in Joliet, a small room with 2 double beds and a bathroom the 4 share. They have at times lived in the boyfriend's car and done without food. The mother has just started a minimum wage job and has no prospects of finding a residence any time soon. We, as life-long friends of the family, would like to take in the child until the mother is on her feet. If the mother is willing, is such an arraignment legal, can we enroll the child in school, apply for badgercare for the child,or will child services attempt to intervene. My wife and myself have been vetted foster parents in the past and have successfully adopted 2 of our previous foster children. We own a home, are stable and have twin 13 year old daughters living with us.