Is it permissible for a mother to fast on behalf of her sick daughter, who broke her fast last Ramadan and whose illness continues until this Ramadan?

It is not permissible for a mother to fast on behalf of her sick daughter or on behalf of any other living person. No one's fast is valid on behalf of another during their lifetime. If the daughter's illness is chronic and incurable, and she is unable to fast, then she must feed a poor person for each day. However, if her illness is one from which recovery is hoped, she should not feed, but rather wait for recovery and then make up the missed fasts.