We are sorry to hear about your loss. There are two kinds of potential financial responsibility: personal responsibility where you would have to pay from your own funds and responsibility on behalf of your mother to pay from her funds. You would only have personal responsibility if you signed a personal guaranty with the facility.
Otherwise, children are not responsible for their parents’ bills, medical or otherwise. You could, however, be required to pay from your mother’s own funds. It sounds like she did not have any funds other than the joint bank account with your father and that account should have passed to your father at your mother’s death, so there is no probate estate.
That said, your father may still owe the nursing home. First, unlike children, spouses are responsible for one another’s maintenance, potentially including health care costs. In addition, what is owed may be your mother’s income that should have been paid to the facility. If that income went to your father, then he would owe it to the facility.