How can I fix a broken shower elbow?
By shower elbow, you mean this part, right?



If so, then to remove any remaining part left in the copper, get yourself some LONG needle nose pliars



Use those to remove any remaining plastic from the copper.

You can buy a replacement at pretty much any Home Depot or Lowe's - so buy one AND some teflon tape



IMPORTANT DO not buy the yellow package - that is for GAS, not water.  Blue and white or red and white packaging only!!!

Wrap 2 or 3 layers of tape around the wall end of the elbow.  Wrap it so that the tape follows the threading - so hold the pipe with the threads near you, start near you and wrap CLOCKWISE around the pipe.  That way as you screw in the pipe it won't peel off the tape.  Not too much tape or you won't get the pipe to fit!

Insert the elbow and turn it COUNTER CLOCKWISE slowly at first, until you feel the threads kind of click into place (avoid cross threading) and then turn it clockwise.  Hand tighten only but make sure it's nice and snug and vertical.

Then reinstall your shower head per mfg instructions.