In questions with a question word, like varför, that word goes first. Then the verb needs to go before the subject. So the first two places in the sentence are spoken for. There are two possible word orders here: Varför svarar du inte? and Varför svarar inte du? The first one is the standard word order, the second one asks why YOU aren't answering.
I'm also not a native English speaker, and for a moment I was wondering the same, so here's what I found.
In short, it seems like some contractions cannot be used in place of the separate words at all times. I think that this is one of those cases.
Maybe the following links could be helpful:
You are thinking that English "Why will you not answer?" is exactly the same as English "Why do you not answer?". I admit they are close, but to me not identical. One implies willpower is in use. I'm unsure if a Swedish version using "won't" would have "vill" as the verb, since Duolingo says vill means "want to", but I've seen an alternate definition for it as "to will, to intend". I wonder at the right way to do that. Maybe "Varför vill svarar inte du?"
This sentence just made me realise that English switches things around too. This sentence can either be: "Why do you not answer?" or "Why don't you answer?" We move the word 'you' over one, mash the words 'do' and 'not', take away the vowel 'o', add an apostrophe ['], and move it back one. This is interesting! Tack så mycket Duo! Does any other language do this? With words such as don't, aren't, can't, couldn't, etc.