Where can I buy a solid graphite stick about 1/8” diameter? I need just a small pieces to fix my broken rod.

tubular glass is better for that.  I think a solid graphite piece would concentrate the stresses so much at its ends that the repair will fail.  Find a piece of a tubular glass rod (graphite will work, but is a little more risky, but it has to be tubular so it forms an external sleeve, not an internal spigot.) to use.  Very good article below.

 

https://www.rodbuilding.org/library/repair-oquinn.html

That’s right. Solid anything is the wrong material for a fix.