Here's how to fix it.

On a butt section, use a heat gun to heat up the butt cap and remove it. Then procede as below.

on anything but the butt section, remove the plug and find a piece of broken rod that you can shove up the butt end and extends out the tip end. Mark that piece at the plug end and remove it. measure 3' down from your mark towards the fat end of your broken piece and cut it off. Use an epoxy like Rod Bond, ProBond, or even 30 minute epoxy from the hardware store and mix up a couple of M&M sized blobs one of resin, one of hardener. Use a toothpick or Q-Tip to put most of that epoxy in the tip end of teh broken piece. Drop your repair part in and use another section of rod or small dowel to push it down. Rotate it to assure full glue contact and seat it nice and snug without spreading your cracks. Clean things up with rubbing alcohol. Let it set, trim to match the section length, and fill the end with epoxy to form a new plug.