Must the mother be informed of her sister's death, given the fear of her health deteriorating, and would we be sinful if we informed her and she died?

If informing the mother of her daughter's death could lead to her own death or permanent disability, based on the doctor's statement, then it is forbidden to inform her of it. This is because whatever leads to the unlawful is unlawful, and a person causing the death or harm of another is prohibited by Islamic law, due to the Almighty's saying: "And whoever kills a believer intentionally, his recompense is Hell, wherein he will abide eternally, and Allah has become angry with him and has cursed him and has prepared for him a great punishment" (Quran 4:93). And the Prophet, peace be upon him, said: "There should be neither harm nor reciprocal harm." One should avoid anything that might lead to her knowing about the death. If she asks about her, she can be told that she is safe, meaning safe from the hardships of this world, for in allusions there is room to avoid lying.