[Zero10]

Shaking the tire will tell you nothing about the intermediate shaft. The gearing in the steering rack itself will be the limiting factor here.
It will however tell you if you have bad tie rod ends or ball joints and you should test them because they need to be tight for the intermediate shaft test to be accurate.

To test the intermediate shaft, turn the car off and park on relatively sticky ground (clean pavement for example). With one pinky finger on each side of the steering wheel (or another finger using light pressure), rock the wheel back and forth. The tires should be staying put properly due to friction with the ground, and without power assist you are feeling any play there is between your steering wheel and the rack. You will need tight tie rod ends and the steering rack mounts will need to be tight for this test to be accurate. You should feel very little play at all. More than 1/64 turn or so means something is loose. You should be able to have a friend repeat this test and trace where the looseness is coming from. Typically the U-joints in the intermediate shaft wear out and the looseness they produce can be really hard to track down.