It sounds pretty obvious that your driving instructor wants to claim because YOUR claim and HIS claim will sound a lot better than just him claiming. Plus I'm not sure how it works, would you get money on an instructir's car's insurance? Do YOU actually get the money? Investigate this first.
Plus, why do you actually need the money? If your back injury really is that severe that you need to pay for operations or anything else, then yes, do claim (if of course you get the money through your instructor). Otherwise, why? To teach her a lesson? Ok, but why should you get the money? Donate it to charity then.
Also, I would get your story straight. As I'm not sure what happened. Had you stopped, or were you still driving when she crashed? Maybe you had stopped too suddenly? I know i did that a lot when i was starting out. What has the instructor said? Has he said you had already stopped, or still driving very slowly? Of course, as she's had an accident before it probably would work in your favour. But still....