Repair frayed electrical wire with strain-relief stopper
Well, it's hard to tell.  That part is called a "strain relief" but in these types of mass produced products, it's often molded onto the cord when manufactured.  Unless the manufacturer sells a replacement part, it will be near impossible to exactly replace.

There is a possibility that you could slide it off that cord, but will be difficult to thread your "slack" cord though it.

Another potential, depending on your Xacto knife skills would be to try to cleanly cut into it from the side, coax it off, and then put it over the new cord, perhaps securing with some wraps of white electrical tape.  The part is held in place by the slot in the cover you removed.

Given the state of the cord, you are doing the right thing, even without the external strain relief, that first cable clamp is doing the internal work.  Your repaired device , even without the external strain relief, would be safer than the current condition.