Cher has tonsillitis bound to be hard to sing and give it her best
A lot of singers have problems with their health...they still go out and nail a performance, even if the performance is slightly below par. What they don't do is completely fall apart and start crying. The cry-baby nonsense may work on 'mummy & daddy', but this is the big wide world of record contracts and making money. If the top bosses can't make money out of you, they will drop you like a bad smell.
Cher's problem I think is that she is very young. When we were 16, would we have had the ability to have reigned back our performance so that we went easy on the throat, but did just enough to get through this round and then fight another day?
As such, she should probably come back next year or seek to get a record contract through other means. It is quite possible that Cowell may take her on.
One thing people must remember is that you don't need to win x-factor to become successful. Conversely, winning x-factor does not guarantee you success. x-factor is merely a show case for your talents. What you choose to do after x-factor is what ultimately determines how successful an artist you will be.