Sounds like you have a great, if not ideal, TOEFL tutoring situation going on: a motivated student, a long time to prepare, weekly lessons.
I 'm jealous!
I don't want to completely avoid your question, but if I were tutoring this young woman, I would think my primary objectives would be the following:
1. Direct student to appropriate TOEFL prep books (which you are doing by asking the question you did). I also liked the Heinle and Heinle one Pink Piggy recommended, although not for very advanced students.
2. Provide student with a significant number of partial and/or complete practice tests, either by recommending a book of tests to buy or giving students copies from your personal test collection, if you have one.
3. Teach general and section-specific test-taking strategies. A good TOEFL prep book should have these.
4. Explain confusing test items
5. Guide student through exercises in TOEFL prep book
6. Informally score practice TWEs
7. Identify student’s specific problem areas, especially within the grammar and writing sections. This can be done by looking at the types of errors made across a couple of tests and also by looking at a small collection of the student’s practice TWEs or other writings.
8. Help with prioritizing: Which things are most and least important? (i.e. Don't worry about the subtle differences between at the beach/on the beach when you can't even structure a basic sentence correctly!)
9. Assign supplementary exercises that target weaknesses, offering explanations where necessary.
10. Direct student to other resources that will improve grammar, reading, listening, writing, vocabulary, etc. and will be helpful for the TOEFL as well as general language development.
11. Assist in developing and managing the student’s long-term test prep and general language learning plan. Teach learning strategies that fit into this plan.
12. Encourage the student to look at the big picture and always relate TOEFL study to general English improvement whenever possible, while still empathizing with the student's need to "pass" this all-important test. Balance the "quick fix" attitude (which is not necessarily bad) with the long-term approach whenever you can.
It seems to me that you cannot effectively help this student with one TOEFL book, not only because each has its own strengths and weaknesses, but because a student will need grammatical information, supplementary exercises, and numerous practice tests that you won’t find in a test prep book. I would pick one or two favorite test prep books to center the lessons around and then supplement with a book of practice tests and a bunch of non-TOEFL resources. These resources could include: Grammar practice books, reading books with skimming, scanning, and timed reading practice, composition books that show the basics of writing an essay, vocabulary books (Essential Words for the TOEFL is one TOEFL based book - it doesn't deal with idioms- but any generic ESL vocabulary books that develop vocab systematically would be good too), idiom and phrasal verb books (Heinle and Heinle's TOEFL book does a great job on idioms for the listening part, but a general idiom book would probably be a good addition), an academic prep listening book that contains talks on academic subjects, etc.. Your student may not want to buy all of these things and neither would you, but she could buy a few of the important ones. And you may have a few materials you’ve collected yourself over the years or a resource library available to you somewhere. You could also see if you could get an examination copy of something and take from that when necessary.
I can understand that students would want a test prep book with the full explanations in the answer key. The Heinle and Heinle one I keep mentioning does this (explanations are brief). But, I don’t really see this as essential when a tutor is involved. It seems to me that
you will be her primary source for explaining difficult test or exercise items. In fact, this will probably be what you spend the most time on during your sessions, assuming she can do the bulk of the actual test-taking and practice exercises on her own time and reserve class time for questions.
Sometimes students will ask about dialogues in the listening section or passages from the reading section, but most often the questions are about grammar. This can be daunting if you don’t feel comfortable explaining English grammar. If that's the case, just do your best. Come up with a series of 3-5 similar examples for an item you ‘re having trouble explaining and try to see if you can figure out what's going on. If you can't explain it, just offer her the examples and tell her to study them. You can also refer her to a grammar book (although there are some types of TOEFL items I know I will probably not find in grammar books), tell her you'll think about it and get back to her, discuss it with fellow teachers etc... Students always want to know WHY, and in many cases I think there is an acceptable answer that will help clarify the issue for them. However, there are times when it needs to be pointed out to students that rules are not sufficient to explain everything. I try to encourage analytical thinking when I teach TOEFL grammar (make them explain why this is the answer and why this one isn’t right), but students also need to see the limitations of rules and explanations and learn to move on when no ready answer is available.
To help your student with grammatical weakness areas, have her buy a good grammar practice book to supplement the grammar material in the test prep book. TOEFL books are notoriously lax on explanations and examples (for a good reason) and your student may also need more practice with a point than the book offers. Good old blue Azar is essential for any student studying for TOEFL and will be useful to her after the test too. Every student should also have a grammar resource book (just info, no practice), like Swan's Practical English Usage. It's essential for teachers too to help explain some of those grammar items that you just can’t find in an ESL textbook.
Good luck! Sounds like a fun assignment!