You did not mention if graphite or bamboo, but, I have removed the broken tip material from the metal tip by the following methods:
I have never broken the tip off a bamboo so not sure if what I have done will work on it or not, but, for graphite if there is any of the graphite tip material exposed, I have gripped it with a pair of needle nose pliers and heated the metal tip top with a lighter and just pulled the graphite out of the tip. It does not take much heat to do this. If no material is exposed, I have taken a very small drill bit and held it with a pair of pliers and just screwed the tip top onto the drill bit and then heated the tip with a lighter and grasp the drill bit and pulled the material out. The above may work on bamboo, but, since a bamboo tip is not hollow, you may be able to use a very small drill bit and screw the tip top by hand onto the drill bit to make a small hole. Then take a very small screw and screw it into the hole you have made and heat the tip top with a lighter and grasp the screw with pliers and pull the bamboo material out.
Just thoughts and ideas that have worked for me....
Fly fishing and fly tying are two things that I do, and when I am doing them, they are the only 2 things I think about. They clear my mind.