Hi Rose,
You definitely came to the right place. To be honest, it doesn't sound like your local repair shop is very good if they can't handle a fairly simple soldering job. I'm not a professional repair person by any means, but even I've done that kind of fix.
It does take a little bit of skill with a soldering iron; you don't have to be an expert by any means, but a bit of experience would come in handy. The part itself apparently has six pins that go through the circuit board (referred to as "through-hole" mounting). The basic idea is you would use a solder vacuum or solder wick to remove the solder from each of the pins in turn, freeing it within it's mounting hole until all six are loose, at which point you can then lift the old part off the board. Here's the part you need.
Asus audio jack connector for UX581GV - Official Asus Partner - Asus Accessories
Now the other part is figuring out how to get to the motherboard. I didn't find any guides, but there's a maintenance manual that at least shows how to open it up.
ASUS ZENBOOK PRO DUO UX581 MAINTENANCE AND SERVICE MANUAL Pdf Download
On the other hand, there is a YouTube video available that does show exactly how to remove the motherboard; that will let you get to the jack to replace it.
ASUS Zenbook Pro Duo UX581 DISASSEMBLY and TEARDOWN - YouTube
If you're not familiar with soldering, I'd suggest picking some broken electronics to practice on before you tackle your motherboard. Once you have a good feel for soldering and unsoldering parts, you can then have confidence in your ability to replace your own headphone jack.