Judging from the youtube videos you can do it as easily as anyone.
The apparent trick is to use piano wire and avoid the heat treating and tempering associated with normal wire.
Measure the thickness of the original spring and compare that to the chart at:
http://www.pianoworld.com/pianowire.htm
One video showed using a strand of steel cable, instead.
Then watch how the wire is wound on the youtube videos.
About all that is needed is a drill, a home made mandrel to wind the wire on and some practice.
To use regular ole' wire, even the heat treating and tempering doesn't sound all that complicated.
The instructions were:
How do you make a small coiled wire spring?
To get the wire to take set you'll need to harden and temper the wire. Start with the hardening, heat to red for a few seconds and quench while it's still hot. The quench is oil, water (brine), or air, depending on the type of steel- if you're not sure use brine (salted water). Then polish the wire to be able to see color again, and temper to dark blue.
But they left out whether this is done on the finished wound spring or on the wire before making the spring shape.
