The volume up, volume down (and the power button) are made the same way:
1- On top: 1 square or rectangular rubbery cover; color orange or clear (in the photo you showed us, they are orange)...
2- ...with a small rubber button in the middle. This rubber button can be clear, part of the rubbery cover; or black, glued on the surface (on that same photo, the button is black). This is the pressure point that clicks on the 2 domes underneath it, causing contact.
3- Underneath the square or rectangular small rubber: a springy square or rectangular metal dome;
4- On bottom: a second metal dome, but if you look closer, you will see that this one is covered with something white inside (underneath) the dome.
If you are missing any of the 4 components, your pressure button will not make contact and will not work.
And this is why your volume up/down buttons were not working. Not because the metallic buttons are defective: the metallic part hardly "breakdown."
But because the buttons of the Headphones Jack and Volume Control cable running underneath the metallic buttons broke down. How do they break down? Due to repeated mechanical action, either the small rubber button (2) is smashed, or detaches; or the whole rubbery cover (1) detaches, letting the springs underneath it (3 and 4) loose (which is what happened in your case, and why the 2 small metallic domes fell out).
This is the guide that you should be following to repair the volume up/down functionality.
You could of course restore the buttons themselves instead of the whole cable, but you will need to scavenge the missing rubbery cover and the 2 metallic domes for each button from somewhere (I usually use other cables with matching buttons). And more often then once, the assembly does not last as long as a new cable. So my advice is to install a new cable but watch out around the headphones jack as it may easily tear the flat cable.