Aux Jack repair

Inside the plug, there is a switch that is supposed to close the circuit when the plug is inserted. The switch has failed (open) and so the system never knows there something plugged in and therefore never allows the Aux to be selected. You need to bypass the switch and bridge the switch pins to make the circuit permanently closed.

I hope that makes sense and helps.

 

^ what he said. When you plug in, the plug shank physically displaces a lever to close a switch and that (usually) grounds a pin that is normally unconnected. When the head unit detects the extra path to ground it knows to enable the aux input.

Replace the jack itself, or replace the entire pcb module, or permanently ground that pin if you don't mind locking the system into aux mode.

There is a small chance you can get the lid off of the jack package and correct the internal switch problem. I've never tried on one of these; only some pcb mount jacks are meant to open up, some are one-piece or welded shut. It would be tricky anyway due to size.

This diagram (generic, scraped from elsewhere) may help:

 

While the failure in time (fit) of a soldering is 5 a rarely used connector has 10 fit and a switch (in the plastic housing of the Jack a switch is included) has even 1500 fit. (Typical values from MIL-HDBK 217)
So probably a problem with the Jack itself. If you'll open the plastic housing, maybe you'll find some dust between the contacts or mechanical parts. But be aware of any possible little spring set free while opening.
You also could try to just blow some compressed air into the Jack at first.

SM-G950F ?

 

I can't swear that circuit works the same as yours, but you're on the right track.