Because that is not the full question, and is technically improper English, as your question would be more properly rendered as: "Which accent does he have?" "What kind of" asks for a general type and expects a broader reply, "which" asks for the specific accent. Think "German accent" vs. "Bavarian accent", "English accent" vs. "Cockney accent", and so on.
It's correctly only if you add "got" at the end, as in "What kind of accent has she got?". I just googled "What kind of accent does he have" (with the quotes) and got 3 290 search results, and then I googled "What kind of accent has he" (with the quotes) and got five search results, and all of them added the "got" at the end. Then I googled the same things with the feminine pronoun and got similar results. I also tried more common questions with an ownership clause and they led me to the same conclusion. The reason for this is the fact that the question requires an auxiliary verb. If you use just one verb, then you'll have a lone auxiliary verb with no main verb for it, which doesn't make sense grammatically.