What is the ruling regarding my sick mother's fasting, who was unable to make up the days of Ramadan from last year due to her illness? Should she give charity instead of fasting, or is it obligatory for her to make up what she missed?

If the mother is temporarily unable to fast, she may break her fast and make up for it after recovery, without needing to pay a fidyah (definition) (expiation). As for making up for a previous Ramadan (definition) that was not completed before the new Ramadan due to illness, it remains an obligation upon her that she must fulfill when she is able, and she is not required to pay expiation for delaying the make-up fast.