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Well, you dont have much choice but to do some cutting and splicing in a NEW MiniJack adaptor (9/16 " where you come up with that one?) its actually called a 3.5mm TRS Mini Jack
In any case, your first task is to remove the broken piece from the laptop.
Secondly, you have to do some cable surgery on the cable that is broken.
YOu have to remove (cut off) the old broken jack, and then attach a new one. Stripping the cable down to the RED and BLACK wires.
There are 4 (really only 3) connections in a 3.5mm TRS minijack cable.
There is the Left Speaker Positive, and the Left Speaker Ground.
And there is a Right speaker Positive, and a Right speaker Ground.
But the 2 grounds can be combined into a single connection. Making it really only 3 connections.
3- 2x Grounds.
Sp;licing is fine....BUT YOU HAVE TO SOLDER, or you are correct......signal degredation will happen
But splicing in a New jack at the end of a cable is quite fine......but again, you MUST use SOLDER. you cant just twist wires together, or even use a crimp.
Solder is the ONLY way to preserve the signal.
So, as long as you solder your connections, signal will be fine.
I cant stress it enough, how important soldering actually is.
Its everything.! LOL
If you take apart a Minijack (remove the plastic ) to see how the cable connects to the jack, you will see it has been crimped....AND THEN SOLDERED on top of the crimp
The crimp will make the connection stronger, and the solder keeps the signal from degrading across the connection.
I make my own cables quite frequently......as i do music production as well.....and im not paying 20 bucks for a simple splitter.....i just make my own.
Anyways....heck you can probably just find an old Minijack on a old cable you dont use anymore (cut it off) and splice it to the end.
Oh man.....that is going to be a challenge.
What i have HEARD has worked in the past,,,,,,,superglue.
and you have to PRECISE with the application of it......if any of it gets around the connection is forever in there.
But the idea is to use the tiniest amoutn of Sglue, and glue the broken piece back to the piece lodged in the laptop.
Wait for it to dry, and then it will just pull it right out.
YOu dont HAVE to use the brokenm cable to glue on the end of the lodged in piece.......toothpicks work great here too.
Use a tiny TINY amount of glue and glue the end of a toothpick to the lodge in piece, wait for dry, and then pull it out.
But again......you must be extremely careful not to get glue ANYWHERE else but on the lodged piece ONLY.