Your's is a tough situation. While I was not faced with the exact same decision as you, my sister died six weeks after my father who was in a nursing home. My family and I often discussed, after the fact, would we have told dad that my sister had died if he had still been alive - we can never come to a conclusion. The answer is - there is no right answer. I know you want someone just to tell you what is the correct thing to do. But, alas, if your decision is made with love for your mother then you will do the right thing - whatever you do. With your mother's advanced Alzheimers, she most likely will not remember on a daily basis about your sister's illness - then what? Do you remind her every day, will she spend the day worrying about your sister, feeling helpless to help her. If you don't tell her, will your Mom sense that there is something that the family is keeping from her.
Go with your heart - no right or wrong - you do not need to feel guilt with whichever decision you make. Be strong - you have a lot on your plate right now!