Considering how dry your picture is, I am guessing soaking the shit out of it in Kano Kroil or Deep Creep wasn't one of your first agenda items.

At this point, you are in over your head, you need a smith.

Pull barrel out of the upper receiver,  and soak the shit out of the spent case with Kroil to begin with.

Tap the back of the case the same way that you would remove a stuck case in a reloading sizer,
Watch this video,
https://www.youtube.com/watch?v=5mVcEvPCWfw

Using a bolt long enough , and plate off the back of the barrel extension face with hole through it for the bolt, thread the bolt down in to the threaded case to remove the spent case that way.  As for tap and bolt size 1/4"x20 pitch, and you want to pre-drill with a #7 for the needed pre-tap channel on a 223 case.


Before installing the barrel back on the upper, scrub the ever loving piss out of the chamber with a chamber brush by hand, then confirm that the chamber walls are smooths as a babies ass.   Hence not chamber reamer grooved/ridges, nor someone tried to polish the chamber and just grooved the chamber to hell that I suspect that the chamber is now and why the spent case did not easily knock out to begin with.


As for running steel coated ammo in the future, make sure to run the upper bearing areas CLP lubed on the wetter. It's not going to stop you from having another spent case stuck in the chamber, but will at least allow you to knock the spent case out of the chamber cleanly with a cleaning rod instead (small trace amounts of CLP transfer to the live rounds just before they load into the chamber, and this trace amount of CLP on the chamber walls assist knocking the spent case off the baby's ass smooth chamber surface.