One of the ways Diabetes can kill is caused by sugar. The Body basically stops or seriously lacks insulin needed to bring sugar (in form of glucose) around your body, so they can be either used or stored in forms of fat.
If they are not used due to Insulin not delivering them around the body, they will travel through (or float through) blood veins and arteries etc. If they meet up with each other they can clog up the blood veins like the same way that fat can. Causing entire limbs to be stopped from being supplied with blood which means no oxygen (from hemoglobin in the blood, which is the main function for the blood veins etc) for the area, which can cause the entire limb to be blue and eventually get necrosis (or quite literally start rotting).
Now, it won't be a problem if there is medical surgeries if these are noticeable and easily rerouted. However, it is fast and deadly if these 'blockages' are caused in the brain or anywhere that is not easily reachable by current medical technology and/or unnoticeable, where I presume the pain will be similar to suffocating or heart attack.