First the model would be helpful, and whether of not the generator is fully working. If not check the 10amp fuse in the generator.
Based on the problems you had with the element prior to it giving out, I would suspect a loose wire somewhere. Have you plugged the cabinet directly into a wall socket to see if the element is working? To do this you will need to use a desktop computer power cord (NEMA5-15P to C13 ) to plug the cabinet into the wall. If the elements get hot and glows red, then the problem is in your generator. Possibly a loose wire in the generator.
If it doesn't glow or get hot, then the problem could be a loose wire in the cabinet, the element itself, the inline fuse, or the temperature sensor. To check the element you will need a multimeter, remove a wire at one end and test the OHMs, you should get a reading of 27 - 31 OHMs, if not you element is bad, and needs to be replace. Be careful, the nuts that attach the wires are very small, and easy to loose, especially if you are working outdoors in such cold weather. You should bring the smoker into a heated area.
If you get a correct OHM reading, then it may be a loose wire, the inline fuse, and/or sensor; which you would need to remove the back of the cabinet. First check the wiring at the plug insert to make sure all those wires are connected, and secured. Next use the multimeter to test for continuity of the inline fuse, and the sensor. If you continuity is bad for the inline fuse or sensor they would need to be replaced. When putting the cabinet back on, be careful not to overtighten the screws; they easily strip the sheet metal.