Well, this is my Cannae all right. With her books and system, I've lost the battle already. So, my question is, dare I ditch her damnable italian textbooks and set her the Hillard and Botting OR the Athenaze (English version, I don't have the Italian version) or do I persist with her damn books which will ensure that she will definitely fail her remedial?
Don't bring your own conceptions about what she should or should not be able to do into this. I.e., thinking you have to plunge right into an intensive course of composition. Or that she's doomed to failure because of the methods of the Italian educational system. It won't help her, and it will only frustrate both of you. What does she want? Is she only interested in passing the exam, or does she genuinely want to understand Greek? This makes a difference. First rule about teaching: don't force your pupil. You have to go at their pace. You can't force them to do something they're not ready for. Particularly in this case.
I know a lot of people here are extremely avid supporters of learning Latin and Greek using more 'natural,' intuitive methods, mastering speaking and writing the language. I'm not criticizing that. I think one learns the language much more thoroughly that way than if one merely learns to translate.
HOWEVER, it is not the only way. What concerns me is the attitude that if you don't learn Greek 'their' way, you're doing it wrong, your knowledge is somehow inferior. I've seen that here far more than I'd like. It is perfectly possible to become competent in Greek using a traditional grammar-based course of instruction. (This is how I learned Greek. While my Greek is not as good as I would like, and I still struggle with composition, I would like to think I'm not completely ignorant of the language.) A rigorous mastery of grammar is not something to be scorned. The problem is when it starts to be treated as an end in itself and not simply a way of accessing the meaning of the text.
Now, as to your problem. You've taken a tremendous task upon yourself. Make sure your expectations are reasonable. How long do you have? You can't expect to be able to make her fluent in Greek in a month, particularly if she already has had negative experiences with the language. What you can do is work on laying the groundwork that she should have gotten in the first place. Not rote memorization of endings, but an understanding of what they mean and how they work. Problem-solving techniques. How to figure out a meaning based on the context. Build up her confidence. I would suggest -- not, mind you, knowing the situation in any detail -- combining progressive readings (something like A Greek Boy at Home, Thrasymachus, or Morice's Easy Stories in Attic Greek) with a course of instruction in grammar. Readings so that she has the experience of reading something and actually understanding it. Grammar so that she can cope with the exam. If you feel composition is essential, make it based on the readings, i.e., have her read a short passage and then write a few similar sentences, changing a verb tense or replacing the subject, object, with another noun.