I am sorry to hear about your mother's recent fall. Falls are one of the main safety concerns for seniors living alone.
It is very difficult to live so far away and it can make family members feel helpless in assisting in care.
Although her fall is so unfortunate, it can be turned into an opportunity to introduce assisted living to your family. An assisted living can be a way for your mother to "step down" to home. You can introduce the idea to your mother and your sister that the assisted living may be a way for your mother to get stronger after rehab and before she returns to home. She would have her own apartment and independence, but the care is there if she needs it. Your sister will probably have peace of mind as well that someone is able to check in on Mom around the clock for the initial time as she gains more strength. In many cases, she would only pay for any care that she would need. This is generally a less expensive way (compared to a nursing facility) and would also introduce assisted living to both of them.
I also advise family members that this is a way to ensure that Mom will be successful when she does go home. You may use this strategy with your sister that this is temporary for Mom but will help to ensure that she doesn't go home before she is ready and risk another fall that could land her permanently in a nursing home.
It would be my hope for you that your Mom and sister might feel differently about assisted living after staying there for a month or so and are able to fully appreciate all the wonderful services (housekeeping, laundry, meals) and amenities (including full time activities director and transportation).
How does your Mom feel about this situation? How does Mom feel about living alone?
Depending on your relationship with your sister, you may want to just gently probe her on whether these are true feelings and fears that she has for her mother or if this is more rooted in her feelings of guilt. It may just take some time to introduce the positives of assisted living (socialization, med management, meals to ensure her nutrition) for her to understand that this does not need to be a negative.
This is such a difficult situation for your family. There are wonderful articles on this site as well as please feel free to contact an advisor in your area to go over more strategies. There are wonderful resources in your area that your local advisor will be aware of that will really benefit you and your family.
I wish you all the best with your Mom.