Hello I believe I can help you with your concern
I am so sorry to hear about your health and about the behavior that your sister has been exhibiting.
It appears that your sister has been exhibiting traits of Narcissistic Personality Disorder (NPD). Individuals that exhibit this type of behavior typically only think about themselves, can easily manipulate others, have a grandiose version of themselves, and repeatedly fail to take responsibility. Here is a link that can provide you with more detail about this disorder
Unfortunately individuals with a diagnosis of NPD typically do not fair well with treatment because they do not believe that there is anything wrong with them and instead they feel that everyone else is wrong.
I think it would be possible to get through to your mother, but you have to try to isolate her away from your sister and have a family intervention of sort to help her see that your sister's behavior has been manipulative and that she has been trying to turn your mother against everyone using lies.
If your mother would agree to family therapy, that would be the best approach and it can help heal your family relationship, but your sister should not be allowed in the family therapy session or she will try to manipulate the session and dominate it. Your sister would need intensive individual psychotherapy before entering family therapy should even be considered.
In addition, this is a good link at extensions of individuals who have NPD and I think your mother may possibly be an extension of your sister's NPD behavior.
Also these books have great information on how to analyze the situation and interact with someone who has NPD.
So far you have been doing a good job at steering clear of your sister because in your fragile health you do not want the added stress of dealing with an individual who may have a diagnosis of NPD. But I think after you read these books, especially the first two, I think you will have enough tools to help you. Also I think consulting with a therapist with experience in treating NPD would be a good idea as well.
In fact if the therapist is willing to sit in on the family intervention of your mother, he/she can provide objective information about NPD behavior that your mother may find useful and help open her eyes.
I see that you are offline right now, but when you get back online I would be very interested in continuing this discussion with you and talking about anything further you would like to share regarding your concern, so if you respond in the chat box I will be able to get back to you as soon as possible.
Thanks so much for your advisement Dr. Z. I am going to spend some time looking at the resource links you have provided me and see what might best suit my situation. I might have a talk with my brother about what you have said and I'll get back to you as to the outcome of that.
Anytime, I am always happy to help. I hope the resources help you a lot and I think talking to your brother would be a great start. My goal is to provide you with excellent service, so if you ever have any further questions or concerns please do not hesitate to contact me at anytime.