What is the ruling of Islamic law regarding the expiation for my mother, who broke her fast in Ramadan for 15 years, some of those years due to illness and others without a valid excuse?

The days your mother missed due to a legitimate Sharia-approved excuse must be made up. However, if she delayed making them up until the next Ramadan began without a valid excuse, then she must make them up and feed a poor person for each day. If the fast was broken without an excuse (due to laziness), then she must repent and make up the fasts, and the passage of time does not waive her obligation to make them up.