From what I can tell, it seems like the Gettier problem comes down to Smith not knowing that the man who has ten coins in his pocket is going to get the job. What about Smith knowing what the successful job-applicant has in his pocket?
I imagine that Smith also doesn't know why or how said man has those coins. Smith doesn't even know whether those coins are there. By stipulation of the problem, of course, he doesn't know who has ten coins in their pocket (this is equivalent to the "what" version of the question, just less "impersonal"). However, it seems as if a slightly different version of this Gettier case would be (epistemically) possibly compatible with Smith not knowing that Jones has ten coins in his pocket, and yet Smith might know half the answer to, "Who has ten coins in his pocket?" (the full answer being, "You and me both," between those two men).
If knowledge has this wh-complement character all across the board, like the concept of knowledge is an elementary function of erotetic logic, then can we frame the Gettier problem in terms of which(!) wh-complements are known in a given case, and which are not, such that we can rank specific Gettier problems by how many wh-complements are defaulted on in any given case? Or does making note of the intersection of the Gettier problem scheme on the one hand, and this interval in erotetic logic on the other, only trivially reinforce the fact(?) that Smith doesn't know that the man with ten coins has in his pocket is going to get the job? As if it would be "pointless" (or not very "productive") to emphasize or otherwise focus on Smith's lack of knowledge-what, knowledge-why, etc., these all being subsumed by Smith's lack of knowledge-that.