There is another way . . .
My parents' model E had the same cordwinder and the spring broke, I think, because it wouldn't retract anymore. On a hunch I bought a shop work light with a cord reel, and believe it or not, the "guts" from it fit perfectly in the original cordwinder's housing. It was a simple swap of all the internals with no dangerous disassembly required. You might look into that. This was in the late '70s/early '80s, but such shop lights are still available. The winding sound changed from a "click-click" to a "zzzzzt" but that was a small price to pay to have a functional cordwinder again.
As an aside, and not to imply that the 'lux cordwinders aren't dangerous to disassemble, you ain't seen nothin' until you've replaced the spring in the motor of a windup phonograph. It goes "sproing" to the tune of about 25 feet of very stiff spring steel, an inch or so wide. Been there, done that, got the t-shirt. Oh, and unlike the springs in the 'lux cordwinder, windup phonograph springs are GREASY because the layers have to be able to slide against each other without binding. It takes "hours" (not really, but what a mess) to get that old dried grease off the hands.
Best of luck to you in fixing your cordwinder.
Joel