Translation:Is she sick?
Before I saw 'ill' was an option, I wondered if Duolingo wanted me to type 'is she disease?' XD
びょう気 is a rather strong word. Before I learned かぜ, I used it with a native speaker to tell her I had a cold. She thought I was confessing to her that I was terminally ill. Needless to say, it was fairly awkward after that, and I only later learned why.
病気( びょうき)is a condition where your health is not normally functioning. So 風邪(かぜ) is also a 病気。But i think, in generally, 風邪をひいた is used when you started to feel sick or have non serious sickness. Consequently the listeners may wonder how bad the sickness is.