I would recommend as a matter of housekeeping that you get your name off your adult daughters account. If you, personally, are ever sued (car accident, identity theft, etc) her entire account would be part of your assets. This can also cause issues for her if you were to die. In this day and age there's no reason to have a joint account unless you're married or in another contractual business relationship For emergencies she can get you an ATM card and tell you her password or you can explore signatory authority if there are longer standing needs that would require you to take money out on a regular basis.
For the medical bills specifically- are you planning to pay the bills (even if it takes longer over a payment plan). If yes, you should be fine; just work it out with the billing departments. If you aren't planning to pay you need to figure out if you guaranteed his payments as a spouse (states are different as are insurance/medical paperwork) and sometimes a spouse is responsible for debt and some debt an individual is responsible for. If just your husband is responsible than your daughters account is safe. However if you aren't planning to pay and you did guarantee payment (or your state says spouse guarantee is explicit) than it's yet another reason to get off your daughters account asap.