You should be able to model the spring as a series of cantilevers, each of which has a concentrated load and a moment at its distal end.
Regarding the 180 degree turns at each fold:
To a first approximation, I would ignore them, just assuming they are rigid enough to transmit the axial force unmolested, and contribute little to the deflection.
To a second approximation, I think you can also model the 180 turns at each fold as a short cantilever, using the developed length of the turn as the cantilever length, i.e., unwrapping the turn for local calculation purposes, and sort of rewrapping the turn, rotating the direction of the force, for combination with the next 'leaf' in order to eventually get the overall deflection.
It's the sort of thing you can do in Excel if you beat on it long enough.
There is another complication; these springs are normally used inside of a rectangular or arcuate cage. I.e., they are not really stable like columns under axial load, but must be constrained by some surrounding structure, so you may have to model the turns as not a pure cantilever, but as a cantilever that may or may not have a support roughly midway along its (developed) length. That may make the force summations more, uh, interesting.
As noted several times already, the shortest/fastest path to getting what you want is to get in contact with someone who is already in the business of making these springs, and, er, develop a commercial relationship with them. Doing otherwise will feed your ego if you succeed, but will also probably cost you years.
Mike Halloran
Pembroke Pines, FL, USA