Must a daughter care for her mother who abandoned her in childhood, especially if the daughter's health does not permit it and her grandmother has other children?

Your grandmother's shortcomings do not permit any shortcomings in honoring her. Rather, she is most in need of it in her old age and weakness. Ask your mother to accept your grandmother's request to live with you if you can do so without harm. Allah has commanded kindness to parents in their old age, as He Almighty says: "And your Lord has decreed that you worship none but Him. And to parents, show goodness. If one of them or both of them attain old age in your life, say not to them a word of disrespect, nor repel them, but address them in terms of honor. And lower to them the wing of humility out of mercy and say, 'My Lord, have mercy upon them as they brought me up when I was small.'"