Your family is not alone in dealing with this type of denial - try calling this number or visiting this web site. They offer free counseling for cancer patients and their families. They may be able to help, or possibly give you a contact in your community of someone who can help.
www.cancercare.org
1-800-813-4673
At some point your mom and you and the rest of the family are going to have to just flat out say that the situation is too much for you all to handle, and that you need to bring in professional caregivers so your mom can get some relief. Your sister's insisting on not bringing in anyone else is grossly unfair to your mom - you may need someone from the outside to intervene to help your sister see that your mom needs help.
Even if your sister does not come around to see reason, please do whatever you need to do to see that appropriate care is available, whether or not she accepts it. You may not ever be able to relieve her desperation - some people do not ever accept that they will lose a battle to cancer - but at least you will have the peace of knowing that you did the best you could under extremely trying circumstances.
There are cases where drs have said there is no hope yet the person has survived. They aren't common, but it does happen. And I would hope that could be the case for your sister - that her belief would prevail. But in the meantime, I really believe you need to reach our for professional help in dealing with this situation.
Sincerely,
Tre