You have a misconception about how it works. Insert the needle. Push the button all the way down, immediately. The clicking sound when you press the button is meaningless, it's just the noise that it makes. You don't count to ten while you're pushing the button; you count to ten AFTER you've pressed the button. It's just to make sure the medication is in, i.e. you're not yanking the needle out too quickly.

The way you're doing it is torture! Trying to slowly press the button is going to make the needle move around and cause pain.

There is a video on their site to demonstrate.
https://www.tymlos.com/taking-tymlos/
Actually, even the video is misleading. Once you press the button, you can let go. It stays depressed. You don't have to keep squeezing it while the needle is in,