Is it permissible to give Zakat al-Fitr to a young, orphaned sister, while the one paying it resides in another country?

The illness due to which you broke your fast has two scenarios: Either recovery is hoped for from it, in which case you must make up the days you missed after recovery without expiation (fidyah (definition)), based on the Almighty's saying: "But whoever among you is ill or on a journey, then [he must make up] an equal number of other days." Or recovery is not hoped for from it, in which case you are not obligated to fast, and you must pay an expiation (fidyah) of feeding one poor person for each day. It is permissible to give the expiation to your sister if she is poor.