What to say to a student who has failed?

I think that you explained the situation very clearly and sensitively in your third paragraph, and that you should send a message to the student along the same lines. All you'd really have to say is that while you of course sympathize with the student, her performance on the final exam makes clear that she did not gain sufficient mastery of the course material to pass the class.

I'll add that I think your message to the student should be clear and concise. The longer your response to the student the more likely it is that you'll wind up in a long, drawn out exchange that stresses both of you out while benefiting no one.

Forget the "tea and sympathy chat," which doesn't do anything to change the situation. What she needs now is some straighforward practical advice about what to do next.

Set out the options she has to move forward from where she now is, and then stop writing. That might be a retake, or a change of course, or even facing up to the fact that not everybody is capable of getting a degree-level qualification - there is no sense creating false hopes by advising her to continue to attempting the impossible, if that really is the case.

If she is insistent about re-grading, you could point out that if you change the criteria for everyone on the course, the end result might not make much difference.

It seems like she had sensible study habits, put in a sensible amount of effort, but still lagged a long way behind the average. You can't (and shouldn't) fix that by tinkering with the assessment criteria. It can be a hard lesson that in real life, you don't get prizes for attempting something and failing, but everybody has to learn it eventually, one way or another.

Actually you can state it pretty much as you have here. You are sorry but her failure here doesn't indicate success in the future in this program.

Alternatively you could re-analyze your own grading procedures, possibly just for the future. Perhaps you are putting too much emphasis on the final exam and making it harder/impossible for students to recover from some common errors.

But that would be for future cases.

In this specific case you could, fairly, see if she can demonstrate the knowledge that you feel the exam indicates is lacking. There are a lot of ways to do that. If she has the knowledge but your exam indicates a false negative you can correct it with a bit of work.

And note that this is not being unfair to the other students. But her begging alone is not sufficient reason to change the grade.