What is the ruling on a husband speaking with his brother's wife with laughter and jesting, and what is the ruling on her consulting him about her private matters and marital problems? And is it permissible for the wife to supplicate against her?

Your husband is not permitted to speak with his brother's wife except out of necessity and within the bounds of Islamic law. Neither the age difference nor her considering him as a brother justifies her speaking, laughing, joking, or touching him. You have done well in advising him, so continue with a kind approach while supplicating for his guidance. You should supplicate for her guidance, not against her, as that might increase her insistence on disobedience, similar to the hadith (definition) prohibiting saying "May Allah disgrace you" to one who has drunk alcohol.