What is the name of the retracting part of a dog lead?
Another example of a mechanism that you may be able to access or modify to your usage idea is contained in a tape measure. There a coiled flat spring steel material is simply anchored to central hub on one end and the other to the tape measure rule itself. These coil springs are setup to work almost the same as the flat coil spring on the inside of a retractable dog leash. In both cases the coil spring is able to provide return pull when the tape rule or leash is pulled out even as much as 25 feet or more.
The coil springs are actually a material that on its own really wants to extend out flat. When coiled inside a confined circular space it strives to straighten which in effect makes it try to expand to the greatest diameter allowed. When the tape rule or leash is extended the coil of this flat spring steel material becomes wound tighter and tighter to the center hub creating more pull and resulting in a smaller diameter to the coil.
The springs like discussed elsewhere here in shades and on garage doors are a different type of spring that is called a torsion spring. These are formed in a tubular manner and when under tension will desire to turn and return to their relaxed position.