http://www.denialville.com/repair-faucet-cartridge/
repairing a faucet cartridge?
I am replacing the cartridge on a single shower faucet, but the old cartridge broke inside the pipe, now I can't remove it to install the new one?
There are a couple things you could try...
A) see if you can get the rest of the cartridge out with a needle-nosed pliers. If you can't,
B) have an assistant turn the water on very briefly. The water pressure should eject the broken cartridge. (Just make sure your not near the faucet when the water gets turned on as the part will be ejected at high speed.)
Good luck!