For my readers, keep this in mind if you are approached to help someone in the family: Offer help only if you don't expect or need the help to be repaid.
At least on the financial front, I have some good news for you. Neither you nor your mother should be worried about coming up with the money to pay the family member's credit card debt. Because your mother receives only Social Security benefits and has no other income, she is considered "judgment-proof." Her income cannot be garnished, with the exception of government debts and child or spousal support. Therefore, should the creditor contact her for payment, you as her power of attorney need only inform the creditor that she receives only Social Security benefits, has no property, is seriously ill and has no intention of repaying the debt. This should stop any further collection efforts.
Acting as your mother's power of attorney means you have the ability to make decisions and act on her behalf. The power of attorney does not in any way make you responsible for any of your mother's debts. The only way you could be held financially responsible for a debt is if you are named as an owner or a co-signer on the account or loan.
Because your family member with the credit card that your mother signed for has declared bankruptcy, the creditor cannot ask him or her for repayment. In the normal course of trying to collect the debt they would likely attempt to contact and collect the debt from your mother. I suggest that you simplify the process and contact your mother's creditor. Explain the situation in writing.
Let them know that your mother is in a position where she no longer makes decisions for herself, and that she does not have income to pay the credit card debt. Be sure to say she does not want to be contacted about the debt again. Once a collector has been informed that they can't initiate contact, the only move they could make would be to go to court. Knowing they will not be able to collect anything from a judgment, they will in all likelihood not waste an attorney's fee and just drop the matter.
Should you be uncomfortable contacting the collector directly yourself, you can always use an attorney to communicate with the creditor. Once your mother has legal counsel, the creditor is required to communicate with the attorney and not directly with anyone else.