Oops! I don't think it should be hard to change a spring. You will need to push the broken piece out from the side it broke on. Peen the end of the new one, and push it in and trim to length. I've had loose ones but not broken ones. If it broke that easily it was probably almost broken to start with.

Replacing a spring can be fairly easy or it can be a pain in the ...

The problem scenario is when the spring has broken off flush with the post (which is often the case) and none of it is sticking out. With that situation, the stub must be removed without damaging the post. The best method is to use special spring removal pliers to push out the stub. The pliers reduce the chance of damaging the post. One jaw is rounded to fit the post while the other jaw has a pointed screw to press the spring through the post. With these, equal pressure is applied to both sides of the post and that reduces the chance of damage. See these:

http://www.musicmedic.com/catalog/products/tool-pl104.html

Without spring removal pliers, a very small punch (roughly the size of the broken spring) must be used to drive out the stub and this method is more difficult because the post must be supported on the back side of the spring to prevent bending or knocking the post out of alignment. It's helpful if you have three hands for this operation: one to hold the flute solidly against a hardwood block for support, one to hold the punch against the broken end of the spring, and another to wield a small hammer to gently strike the punch.