"Is she sick?"
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.
There are several options, depending on level of familiarity or tone you want to convey.
- 彼女、病気？(very familiar, fairly neutral/distant tone)
- 彼女、病気なの？(very familiar, somewhat softer/more concerned than the above)
- 彼女、病気か？(somewhat familiar, very gruff/standoff-ish tone, along the lines of "what's her problem, she sick or something?")
- 彼女、ちょっと具合悪そうだけど... (somewhat familiar, actually a statement that she seems a little unwell but can imply the same question, somewhat concerned yet distant tone)
Disclaimer: Of course your actual tone of voice/cadence will dictate how these options are perceived, but I've tried to give a description of how they are typically/commonly used.