I also thought this usage of (his) flirt was strange. As you noted, a flirt is typically the person who flirts.
After a moment, I thought there were a few interpretations. The first one was that flirt is being used to mean the target of flirtation, the person who is being flirted with (= Nimue). I feel like this kind of inversion is uncommon, but not unprecedented. I can't quite think of a different example though, but I'm willing to credit the writer with taking creative liberties here.
Then I would say turn his flirt is a bit of wordplay. Notice that she literally turns to acknowledge him. But turn someone can roughly mean persuade someone.
verb (used with object)
28 to persuade (a person) to change or reorder the course of his or her life.
6 e : to cause to defect to another side
By the end of the exchange, he persuades her to become friendly.
If instead we assume flirt means flirtation, then another possible interpretation/word play is that turn can also mean transform/convert:
6 b(1) : convert, transform
// turn defeat into victory
You could say he converted his flirtation from a failed invitation into a friendly introduction ("defeat into victory").
Of course, it could also be a mistake. All in all, I wouldn't recommend this kind of usage. For clarity, I would simply stick to the usual meaning of person who habitually flirts.