Broken hips are almost the kiss of death for old people. A doc told me my aunt was likely to pass within a year after her hip fracture. She made it three years.
So sorry about your mother. Hospitals and elderly pneumonia seem to be common. All those germs and so little resistance in old age. All I can tell you is that I will never allow my mother to die in a hospital. So there.
This is probably too personal . . .but why did you not go see her?