Your body does most of its healing while you are in deep sleep. So if you did not sleep it would take a lot longer for your body to heal. If you did not sleep for a week you would be in rough shape. Your immune system would not work as well and you would not have very much energy. It would be hard for you to concentrate and you would have trouble remembering things. Assuming you did not die from a heart attack or get sick and die because of an impaired immune system and stress I think you would still be alive.
It would probably take 2 weeks or more to die from a lack of sleep. However they would pass out long before they died from a lack of sleep. The person is not going to be able to function properly. If other people were not around to protect them the person could accidentally die or die of starvation. A person that goes without sleep for a long time is going to turn into a crazy person with very little energy. They could easily walk in front of traffic or something.