Can I sue the county jail after being attacked by another detainee and having my nose broke?

State laws vary significantly, and I am not in your state, but you would retain a local civil rights lawyer who would need to prove the officials knew or reasonably should have known that you would be harmed or that there was a substantial risk that you would be harmed and the officials did not act to prevent it.

I agree with Mr. Lassen. You need to review this situation in detail with a civil rights attorney in Illinois for some serious evaluation and advice about whether you have a case here. I noticed your comment in response to Mr. Lassen's answer. This is a real-life legal situation, not a bar bet. No lawyer worth anything is going to give you an off-the-cuff opinion and the kind of general proposition you asked about is meaningless. Legal controversies do not depend on broad statements of principal. See a good civil rights lawyer about this.

Find a civil rights attorney and speak with that counsel in person on potential merits of your case.

Failure to protect cases are difficult to prove. The fact that no officer was present helps. Is this against Cook County? Was there a history between you and the other inmate? Was the jail and/or guards aware of this? You will have to prove that the correctional officers were more than negligent. You will have to prove deliberate indifference.