Is it permissible to consider a loaned sum of money that has not been returned as a discharge for another sum of money of unknown source and owner that is in the lender's possession?

This amount does not suffice for your debt because the debtor still believes he is in debt, and because this money, which you found without knowing its source or owners, is treated like lost property that should be spent for the general welfare of Muslims. Ibn Hajar al-Haytami mentioned that if the owner of lost property is unknown, it should be spent on the public treasury's expenditures. If you consider this amount as a debt owed to you, then you have benefited from it yourself and have not spent it according to its legal channels.