Mother gave sister POA, but sister has become ill and can't help to care for them. Any advice?

Only your mother can assign a POA. Your sister can't assign her POA to you.

You should have POA. If mom is still competent mentally she needs to change over the POA to you. If she is incompetent you may need to pursue guardianship.

Often a POA will name a secondary or back-up if the primary can't perform their duties, have you checked the actual document to see what it says?

Check the specific wording to see if it allows your sister sole or primary responsibilities, and/or if you're named as an alternate.

Is your sister's illness short term, or does it appear as though it's of longer duration and likely to prevent her from handling responsibilities for your parent?

Also, you write that "your mother" apparently executed the POA, but then you refer to "them". Are both parents still alive? Is there any issue with mental cognizance that would prevent them from creating a new POA?