Flute springs may seem a bit "deceiving" in their name since they aren't spiral-type springs that you would find in something like a ballpoint pen. Instead, they are actually more straight, although they will have some curve to create tension. They are also very, very small in terms of diameter, which means it takes a lot of skill and expertise to install them. We recently caught up with flute finisher Matt Keller, who explained the process for us.
The hole where the spring is inserted is just under the post. We tried to enlarge the area so that you can see it in the photo below. Take a look at your flute, and you should be able to find it.