Flexible mates in SW
Here is my answer! An assembly in SolidWorks 2012. I hope you can open it.
I try and see if I can make a proper tutorial on it in this answer on a later time. In the meantime, you can have a look at my solution.
The flexible cone I made was first made as a new part, where I constructed a simple circle sketch. This sketch was then mated in the assembly with the wheel (concentric), and the plane on which the circle was draw, was mated with the side of the wheel. Now the cone can turn around. Then, I mated the right plane of the axle with the corresponding plane of the cone. After that, I made a line sketch on the right plane of the axle, with its point ending in the point of rotation of the axle to the wheel. This end point was then mated to that midpoint of the wheel, so the axle movements are defined. After that, I made some sketches in the cone part while being in the assembly. The result is a loft with bobbles, which looks like the hub cone you showed (a bit, at least), and when the angle of the axle in relation to the wheel is changed and you REBUILD the assembly, you will see the cone has changed shape, so it still visually connects the wheel ad the axle. I must say that the sketch used to define the positions of the planes defining the loft, is rather complicated. Please, take a look at it.
NOTE: The standard position of the axle to the wheel is perpendicular to the wheel. DO NOT bend past 40°, or planes inside the cone part start to permanently shift around, which causes the thing to look a fair bit less appealing. If the picture you added to your question depicts the spring in full extension (i.e.: the axle makes it maximum angle with the wheel), then I think the solution from my assembly will work.
Have fun with it!