Looking for a data set that shows hospital patients' illnesses and their vital signs, during their stay
I'm assuming you are looking for medical datasets, not only what you have stated in the question, but also some image recognition datasets which need something like CNN.

Try this one. This dataset contains several medical features including blood sugar, serum cholesterol etc, and wants you to find out the presence of heart disease.

In the meanwhile, there are some medical competitions and datasets on Kaggle, including the famous Data Science Bowl.