How can I pay off my deceased mother’s debt despite not knowing the whereabouts of the creditor, and my father’s repeated refusal to pay it off even though I have given him money for that purpose?

If you do not know the whereabouts of the debt's owner, ask your father about him. If you know his whereabouts, then pay the debt yourself or send someone to pay it. In any case, you are doing a good deed, whether by paying your mother's debt or by helping your needy father. Children are not obligated to pay their mother's debt from their own money if she did not leave any inheritance, but it is commendable for those who are able, out of righteousness and benevolence.