Don,
Attached is a link to a YT video featuring the cord replacement of an Electrolux Corporation- branded cordwinder. I believe Cordomatic Corporation manufactured both it and Eureka's Cordaways, so the design is somewhat similar albeit older than yours.
A bit of history ... the Cordomatic Division of the Vacuum Cleaner Corporation of America made its first retractable cord reel in 1939. The company later became Cordomatic Corporation. Eureka's first use of its Cordaway was in 1965 on several canister models, including the new Model 1100 Empress. Shortly afterward it introduced its first Cordaway upright Model 265.
Some suggestions:
1. Wear goggles & heavy gloves in case the springs tension let's loose unexpectedly.
2. Set the locking mechanism prior to removing the Cordaway cover to hold the tension.
Regards