Following old directions found online I opened an Electrolux Ambassador III to fix the cord winder. When reassembling, I found a loose white wire. Where should it be reconnected?

Did the motor run before you worked on the cord winder? If so, and the white wire is disconnected on one end only, then I suspect that the loose end went to a connector on the inner end of the cord winder to couple power into the motor. The winder most likely has a pair of slip rings that couple power from the cord, which turns on a drum, to the backing plate that doesn't turn. Guessing that the winder uses a pair of brushes to couple power into the slip rings from the cord wires which then distribute power to the switch and motor windings. Good luck! Did the motor run before you worked on the cord winder? If so, and the white wire is disconnected on one end only, then I suspect that the loose end went to a connector on the inner end of the cord winder to couple power into the motor. The winder most likely has a pair of slip rings that couple power from the cord, which turns on a drum, to the backing plate that doesn't turn. Guessing that the winder uses a pair of brushes to couple power into the slip rings from the cord wires which then distribute power to the switch and motor windings. Good luck!