Can I sue the hospital for my uncle who has dementia; fell in his room; broke leg?

If your uncle is a "disabled person" under the law and not able to handle his own affairs, a guardian would have to be appointed by a court, on a petition to adjudicate a disable person, over his person to bring a lawsuit. This assumes your uncle made no written direction before he became ill to appoint an agent holding his power of attorney. The guardian appointed could then stand in the shoes of your uncle. In that case the guardian appointed could investigate with a lawyer whether or not a case were warranted with the facts. Consider also that a hospital would potentially be the defendant and therefore, the Illinois medical malpractice statute would most likely be applicable.