Hi
@Jakesterblack
I have a different perspective. I wouldn't make yourself less available in hopes that your mom would "get it". This would require her to have really good insight, reasoning, and rationalization, and these cognitive skills become impaired throughout the Alzheimer's journey. Anytime we try to reason with a person with Alzheimer's, we may simply be wasting our time- this is not the language that people with Alzheimer's speak anymore. This also asks her to make inferences about your situations, and that is quite abstract for a person with dementia.
A big question with any type of dementia is capacity. Capacity is an ability to understand and appreciate information well enough to make sound choices. Capacity fluctuates, so your mom might make solid decisions sometimes, and not others. However, very complex decisions require a lot of cognitive ability, and your mom might not have this ability any longer.
So, while we always respect her decision, the question is: when you ask her if she would like a cleaner, is she capable of making a good decision in response?
Perhaps it is not a matter of "realizing" that your sister is sick, but truly not being able to understand it because of brain changes.
Do she have the ability to understand how sick your sister is?
Does she have the ability to understand the strain you are under?
Will her brain allow her to understand these things?
Is it fair to ask her to understand these things?
When people lose the ability to have good understanding, insight, and appreciation for choices, then we need to step in and make these decisions on her behalf.
We certainly don't want to make her uncomfortable. However, in the most gentle and supported transition possible, introduce a carer or cleaner. Perhaps a few "friendly visits" first with you and your sister; and then maybe once per week a short visit, and slowly increasing the time. Ensure the hired person has a chance to know your mom well- give them lots of information about her life, preferences, hobbies, interests... so that they can support her for her unique self.
Just my two cents, hope it helps
Kim