What is the ruling of the Sharia on attempting to convince a girl's family to marry her after the engagement has been broken, and is it permissible to speak with her to learn how to convince her family?

There is no harm in striving to convince your fiancée's family to involve righteous people as mediators. If they agree, then praise be to Allah. If they do not agree, then know that Allah has averted from you something that contained evil for you, for Allah Almighty says: "And it may be that you dislike a thing which is good for you and that you like a thing which is bad for you. Allah knows but you do not know." (Quran 2:216) Women of faith and good character are many, and if this matter is not finalized, then seek another. There is no harm in addressing her regarding the engagement, provided there is no soft speech, seclusion, or impermissible gaze.