If you can remove the handle from the cable, I have had success with this: purchase some clear plastic tubing (18") that will slice tightly over the end of the cable sheathing, or screw over the threads if there are threads. Tighten with a cable clamp. Wire tie the cable in a vertical position. Fill the tube with a light oil. Give it a couple of days. The oil should wick down between the cable and the sheath. If you can get the cable to move, work it back and forth carefully. It will be lubed for a long time.