"Is she sick?"
Why can't we just get furigana? In parentheses even, if it's too hard to implement. I have never seen びょう気 written anywhere, in my seven years of living in Japan. Either go all kana, or use furigana if you're going to use kanji at all.
I favour a toggle system where I just tap the kanji to see them "roll over" into kana.
This is a common practice when part of the word's kanji hasnt been learned yet, but it would be great if we got to learn more kanji!
That would imply a question of whether her illness is the reason behind something. 〜なんです is used to indicate the /reason/ behind an action or event. "Why isnt she here today? Is she sick?" Might be a conversation where you'd hear 「病気なんでねか？」
No, I think "why is she sick" would be 「かのじょはなんで病気になったんですか」(lit. Why did she become sick). OP wrote 「なんですか」at the end (which is kind of hard to translate, but adds a tone of surprise/incredulity to the question I think), rather than「なんでですか」which means "why is that?"
At OP, I think your translation is also correct, but I don't think it's necessarily more correct; it just depends on the context and relationship between the speaker and listener. It feels somewhat more colloquial than what Duo has taught until now though.