Clara, this is an interesting one.
What I sense from you is your reluctance to let her win, but at the moment nobody is winning. Besides, when it becomes a competition (or a war) then you have already lost. What you need is a different way of seeing it (both of you) so it no longer can be considered competition, it becomes you and she working together to find a solution.
You're a smart lady. You have tertiary qualifications. it stands to reason then, that she has inherited some of your brains, at least. This will only make it harder to battle with her. You MUST avoid this.
To deal with the ODD side of things, and to help you learn how to negotiate more effectively with her, read "The Explosive Child" by Ross Greene. The concept there will suit kids with a range of problems, it's got quite broad application because it's so easy to tailor it down to each individual. But the first and foremost rule - pick your battles. Avoid them where possible, but sometimes you can't avoid. And where you can't avoid, there are ways to get what you want without it being a competition. The next rule - let her see that you are no longer competing, you are her support. Again, the book's methods show how this develops, you really don't need to do anything extra for that one if you're already following the CPS system. (CPS = Collaborative Problem Solving).
Once you've got a handle on dealing with her in this different way, the next step is much easier. Discussing home schooling - I think you need to sit down with her and work out some rules between you.
Here are some starters:
1) She needs an education. The law says this must be provided. Why does she not want to go to school? Is this insurmountable for her? Why? Can anything else be done to sort out what has upset her? If not, then don't push it. But you need to know, so you can at least understand and be sympathetic. You both need to be able to communicate in a positive way, even if you don't always agree. It's OK to agree to differ.
2) School work must be studied in school hours, at least. Home schooling is not automatically a licence to play all day. If she so desperately wants to be home-schooled, she has to agree to a number of things:
i) work during school hours
ii) do the utmost to complete the required amount; more if possible
iii) cooperate, no emotional blackmail, no whining - SHE is the one who suffers if the work doesn't get done because she's stalling or trying to get out of it.
3) The best thing to aim for is a good study and work ethic. She will develop her own unique methods which may well be very different to what you expect - give her some leeway in this, as long as the work gets done. For example, we're supposed to have difficult child 3 on a timetable - science for an hour, then history for an hour, then english for an hour - but he doesn't work that way. Instead, he does all his science work for the week, in one session. Then he moves to the next subject. He works in different locations, often on the floor. It's been cold here the last few days so he's done a lot of work still in his pyjamas, snuggled up in bed. But the work has been done well. He's comfortable and can concentrate.
4) We set a quantity of work that needs to be completed. We do vary this a bit because sometimes he just has a bad day, especially if our routine is disrupted (I have to go out, or visitors drop by, or I get a lot of phone calls). But difficult child 3 knows that if he has not completed all his work, it weighs on him and he will do some over the weekend or other times outside school hours.
We put in place a number of different ways in which to learn the material. Computers are great, for us. difficult child 3 has just completed a unit of astronomy which was almost entirely on a DVD-ROM. He's registered online with Mathletics, an international Maths coaching website which can be streamlined to the child's abilities. He has some subjects which are more challenging for him - I keep an eye on how his week is going and recommend him doing these on days when he has my undivided attention and no interruptions. We do NOT do these topics on days when we go shopping, for example. On those days he gets very little done, but we do bring his easier subjects with us such as maths & science.
difficult child 3 is a very visual learner, so having large quantities of printed material as worksheets stands him in good stead. We have a correspondence school to look after him, but he still needs me there as support and supervisor. But it does mean I don't have to write the curriculum myself.
Let your daughter know what sacrifices you'll have to make to do this with her. Make it clear that it won't be easy.
But before you begin anything, read "The Explosive Child" or you won't be equipped to cope with the clashes between you.
difficult child 3 & I have got his schooling to the point where we really don't clash any more, except in extreme circumstances (very rare, we try to avoid it). As a result he's working well and is very motivated. This has been a joint effort to get him here, but I now see him doing so much better than he ever did in mainstream.
Good luck!
Marg