1. There is no legal definition of next of kin. It is usually taken to be a member of the family. When your mother went into the home they should have asked, for their records, who the next of kin was. This question is usually asked of the person concerned, but if your mother was mentally incompetent when she wnet into the home then you - as the person who arranged for her to go there - should have been asked. If this was not done, then make sure the home records you as the person aith P of A, & as next of kin. If necessary, explain the problem with your sister to them.
A person who is mentally incompetent cannot make a valid will. If this was done, then you would (after your mother's death) have to dispute any such will using as evidence the registered P of A - which will be dated some time before the will.