Is it sufficient to give charity—feeding a poor person for each day—in lieu of making up eight days of fasting for a person who broke their fast due to illness and was unable to make them up?

The one who was unable to fast Ramadan due to a valid excuse (such as illness or travel) is obligated to make up the missed days. If they are unable to make them up until the next Ramadan arrives, there is no sin upon them, and they are only required to make up the fast. However, if they were able to make up the fast but did not do so until the next Ramadan began, they are obligated to make up the fast along with feeding a poor person for each day. Therefore, the person mentioned in the question must make up the eight days without feeding, because they were unable to fast until the following Ramadan.