LillyCo... From what you have said here, your Mom already needs more help than she is receiving. She probably thinks she is fine at home alone but she also thinks she has not told you that same story 5 minutes ago. She may believe she can manage just fine but she also thought she was appropriately dressed when she arrived. Your sister is in denial. Who wants to believe mom has dementia, but there comes a point you have to accept the diagnosis you are given. Many doctors are CLUELESS when it comes to dementia. There is no medication available that will "improve" Mom's dementia. The medications available do not alter the course of the disease, do not make it better, nor slow down the progression. In a small percentage of people it makes the remaining brain connections work a little better. But this is not long term. Dementia marches on despite the best efforts.
You have the responsibility. Mom gave that to you with the POAs. Apparently, hearing of your sister's denial, Mom had good reason to do what she did. I understand being young, starting a family, and having a live far away. But Mom is unable to maintain and is unable to obtain or even allow the necessary help at this point. It is up to you. Yes, you are sitting there waiting for the next crisis that is going to force your hand. If you do that then you will not have the time to search out the right responses and places that Mom needs. You are worried about sister being angry. That she does not agree with you. Sometimes you just have to leave those with denial behind until they can catch up.
You need to go to your Mom, make an appointment with a dementia specialist (then can't be fooled as easily), go with her and inform the doctor of your observation, and get an appropriate diagnosis. Then you need to make the hard decisions that will get Mom the care she needs. Sister being mad and Mom being upset for a while is better than something horrible happening to Mom... and that is a possibility. I have been there done that and my only regret is that I did not do it sooner
Hang in there and know you are only trying to do what is best for Mom... and go do it!