Reddit mentions of Desk Grommet 2 Inch, Plastic Desk Cord Cable Hole Cover Grommet - 10 Pack, Black

If you have a wooden desk, the Lepai amp has mounting hardware which can be used to screw the amp into the underside of the desk. It can be mounted towards the desktop's edge towards you off to the side so you can access the dials. Then, most of the speaker wiring will be under the desk and will come over the back or through a grommet to the desktop only where the speakers are.
The only thing is the amp will be upside down. I have that Lepai amp mounted sideways on a wall in my office. It's not that distracting. You just turn dials toward 10 to turn up the treble or bass.
For the headphones, I would suggest a USB DAC to have the headphone always hooked up, so you don't have to plug it in. Using a USB DAC makes the headphone/mic a separate output/input source. Then, you can toggle audio outputs and inputs using the windows speaker icon, or use Voicemeeter Banana or some other mixer, whether physical or virtual. I really like voicemeeter - useful for muting idiots on Discord and having both speaker and headphone on, etc.
Here are some cheap cable management things to help manage the cables under the desk:

1. cable clips or these cable clips
   
2. grommet drilling supplies if you have a wooden desk
   
3. grommets
   
4. Toilet Paper Holder screw one of these under the desktop and hang the always-plugged in headphone there.
   

Good luck. Just occurred to me that you may want to consider a "desk Grommet" like this. I would wait until you're done with the hole and see how it looks. If your edges end up looking messy or you see a bunch of fiberboard, this will hide the edges:
https://www.amazon.com/Desk-Grommet-Plastic-Cable-Cover/dp/B076HJH2WQ/ref=sr_1_3?s=industrial&ie=UTF8&qid=1550704465&sr=1-3&keywords=desk+grommet
https://www.amazon.com/Desk-Grommet-Plastic-Cable-Cover/dp/B076HJH2WQ/ref=sr_1_3?ie=UTF8&qid=1550704138&sr=8-3&keywords=desk+grommet