Am I permitted to advise my cousin to seek a divorce, knowing all that has transpired between her and her husband in terms of problems over the past twenty-five years?

You should advise your cousin for her own good, for originally, a woman is forbidden from seeking divorce or khul' except when there is mistreatment between the spouses, or when the wife intensely dislikes her husband to the point that she cannot fulfill his rights. In such a case, she is permitted to seek khul' at that time, based on the Almighty's saying: "{But if you fear that they will not observe the limits of Allah, then there is no blame upon them for what she gives to free herself}" [Al-Baqarah: 229].

Therefore, you should advise her to strive to reconcile with her husband. If he improves, that is good. Otherwise, she should weigh the harms of divorce or khul' against the harms of remaining with him, and choose the lesser of the two harms.