Bear in mind that the story is written from the perspective of the narrator. It's entirely possible that his belief that he is a half-elf is implied to be true within the story by mentioning how natural the blade felt in his hand.

Or it could be that Gilead's Blood was written without certain elements of background being made entirely clear to the authors. There are no half-elves in official Warhammer material, only those who think they might be half-elves.