What sudden illness results from too much or too little sugar in the person's blood?

Hypoglycemia (from too little sugar or too much insulin) is an emergency causing unconsciousness, coma, brain damage and death. It can usually be treated at home with sugar and food.
Diabetic Ketoacidosis (from too little insulin) is an emergency causing lethargy, dehydration, fever, blood acidity, breathing trouble, coma, and death. Once advanced, it can only be treated in a hospital using specific fluids, insulin, and monitoring. It's not caused by too much sugar in the blood though you will generally see that at the same time.

Hyperglycemia (too much sugar in the blood) is deadly in the long term, causing blindness, gangrene in limbs, heart and kidney disease, but it's not a "sudden illness" since it takes months or years to cause the worst side effects.