Will the family be sinful if they do not marry off their sister who is mentally ill, in order to avoid spreading news of her illness or her inability to bear the responsibility of marriage?

Preventing a sister from marrying when she is in need of it is forbidden by Islamic law and is considered "al-adal" (undue obstruction). However, if she herself refuses, then there is no blame upon you. Marriage may benefit her by fending off the harm of lust, protecting her from immorality, securing the dowry and maintenance, attaining chastity, and preserving honor. She should be married if medical professionals state that her ailment will be cured through it.