When he dies, his estate will be responsible to pay his debts.
His debts would typically include final medical expenses, burial expenses, final income taxes, and then any other debts like credit cards, personal loans, mortgages, car loans, etc.
Leftovers, if anything, would go according to his will if he has one or state intestacy laws if he doesn't. If there are still debts left over after all assets have been liquidated, then those typically go unpaid and she is not responsible for them.
Determining what is in "his" estate and what are "his" debts is a matter of state law, so I'd recommend she get a quick consult with a Michigan attorney. Almost certainly (ETA: Maybe not, see post #11 below) the credit cards are "his" debts, but what of "their" assets would end up in "his" estate is where she would have to look at state law and their specific facts and circumstances.