Is it permissible to break off an engagement to a girl afflicted by jinn, given her family's refusal to send her to a lawful healer?

It is permissible for the questioner to seek legal ruqyah for his fiancée, because the Shariah commanded seeking treatment, for every disease there is a cure that Allah has sent down. One must beware of sorcerers, charlatans, and soothsayers. The Prophet, peace and blessings be upon him, forbade going to them and believing them. If the girl's family refuses to send her to sorcerers and charlatans, then this is their right and duty, and you should encourage them. However, if they object to legal ruqyah, they do not have the right to do so, because treatment with ruqyah is permissible. If she is your wife, then your word is final. But if you are engaged, you only have the right to advise and guide.