Jax, this is another example of how we are all so caught up in our own lives that we fail to remember that everyone around us is also going thru tough times. I know this is hard for you. I understand how you have to feel so torn every day about what you can or should be doing. I'm sure that your sister appreciates your "heart" in this.
My experience in working with Alzheimer's patients in the past in the nursing homes and then having been the primary caregiver for my friend for 4 years is that Alzheimer's people function MUCH better when their lives are not disrupted. They thrive on familiar. That includes familiar caregivers. They may not recognize these caregivers or even know their names. But they respond to them better than they do to a new person coming in. So, as hard as it is for your sister, the situation she is in is probably the best for your mom. Maybe the thing that you can do that will be the most help for your sister would be for you to offer her and her family as much support as you can possibly give her.
Some suggestions: (and take them all at face value. They're ONLY suggestions)
* You and your brothers could pay for an alarm system to be installed in your sister's home so that she could get a good night's sleep without having to worry about whether your mom is up and wandering out the door.
* Offer to pay for someone to come in at least once or twice a week in the evening so your sister can get out and do what SHE wants to do. You said she has someone during the day, right? I assume your sister is working a job and that's why she needs the daycare?
* Have your sister's kids come out to visit you and your family for a week or two. That in itself might give your sister a big break.
* Even if you don't do anything else, at the very least, send your sister flowers or candy or something every once in a while just to tell her how much you appreciate the sacrifice she is making to take such good care of Mom. Let her know that you are grateful. That in itself will go a long way.
You may not be able to do alot for your mom right now. But do what you can for your sister to make HER life easier. The more you do for your sister, the more your MOM will benefit
I'll be praying for you, brother. I understand. Really, I do. Alzheimer's a CRUEL disease.