Since you don't hear the drive even "try" to eject, the problem could simply be the front panel of the case. That big eject button in the images simply pushes on the drive's built-in button when pressed. This "soft clicking" sound you describe may indicate that the button is broken.
You should remove the drive from its caddy and try to eject the disk by pressing on the actual button of the drive unit, which is most likely still fine. If you can't remove it from inside the case, you should be able to remove the front panel of the case. There is usually a large gap at the bottom, large enough to grab firmly and pull out (might require some force).