Mom was told by her youngest of three daughters that she was being admitted to rehab for therapy (for walking, hip problems). Mom expected to return home. Sister #2 (who resided at MOM'S house with her) is now refusing her return because she doesn't want to assist/care for Mom . The nursing home states that Mom needs 24/7 care and that she can't take care of herself to the extent that she believes she can. Mom's limited mobility is at the heart of the issue (no actual "illnesses"). Mom is securing an ombudsman to facilitate her leaving the nursing home without a recommendation from the nursing home. Then what?
Can you contact the VA and ask to speak with a social worker? I would also try to get the NH social workers going by helping you with solutions. This is a complex situation, and you're right to refuse to care for him, but it may require intervention by APS to find a place that will take him, if he agrees to assign his SS payments.
Does your mother have the resources to get assistance at home? Have you looked into what resources beyond your family are available?
As a practical matter, it sounds like you and your sisters need to work together to take care of your mom, since it sounds like #2 needs help. There are people who can help you facilitate a conversation if you can't work it out among yourselves.