Here is a link to the breakdown of the laptop, Asus K53E-MT1 Motherboard Replacement that headphone jack is on that separate board with the USB ports on the right hand side apart from the motherboard, which is good news. The hard part about this repair however is going to be sourcing that part to replace it.
I've got the same laptop, Its always been a good machine for what I do with it except I had to replace the charge port connector once but thats a common issue not just with that model laptop, but all of the laptops with that connector. You might have a look at the center pin while you've already got it torn down, thats where they always break and sometimes can be "repaired" by adding solder to support it. Its not a correct fix but a get-by if it's started to crack