Well in German we say "mögen" oder "wollen" but in the last generations there was an order to be extremely polite and never say "ich will" because it sounds very egoistic. So some people never use "wollen" in German (just if it would be correct) but at each time "möchten" what is closer to the meaning of "to like". So this people would never translate to "Das Kind will nicht schlafen". I think that is the reason for this constellation happened here.
like= mögen (from: https://www.dict.cc/deutsch-englisch/m%C3%B6gen.html) want= wollen (from the will one has inside)
Saying you WANT something sounds childish and impolite, so if grown-ups WANT something, they say they "would like" something. At least that is polite English. So "wanting" and "liking" are just shades of politeness. So both verbs should be both accepted also in the german translations. But children are just NOT polite. And DUOLINGO can be quite childish :)
That's bad english to be honest. The sentence sounds bad. You can use it in some specific scenario like: "I don't want to sleep", "Then go", "No, I said I would NOT like to sleep". But this is a specific scenario and accepting it, I think, wouldn't be weird. Rather, you are supposed to say "The child would like to not sleep" and I'm pretty sure this works...