Because "что" is not equal to "what", it has a little bit narrower meaning, it can't be used when you talk about object attributes, for example. So "какой" has all the meanings of the word that "что" doesn' have. Also "which" is rather "который", but we use "какой" pretty often as well. Hope it explanes something :)
It might be clear if you translated the sentence more literally: "Which/what last name does the writer have?"
In reverse, you wouldn't use какой to ask "What are you eating?" = что вы едите?, because it's not attached to anything in particular, but "what food are you eating?" = Какую еду ты ешь?
Using "writer's name" is much more natural English than "name of the writer", which is more poetic, formal, or stylized ("stylized" means "out of the ordinary", done in an unusual way). It can be used to good effect if you want to lend artistic weight to what you're saying. For example, there's a movie titled, "In the Name of the Father". That would sound a lot less imposing and weighty or profound if it were "In the Father's Name".
Another thing: Duo's Russian course uses the Russian word Это and it's English translation "this" far, far more frequently than you would encounter it in American or British English. Most of the time, you'd see 'the writer", not "this writer". But since "the" doesn't exist in Russian, I'm not exactly sure what's going on with the prevalence of это/this. I wonder if Russians actually use it so much, because it seems like a substitute for "the", so why don't they just go ahead an invent a word for "the", since Duo is indicating that "this" is used like "the" so much.