That broken contact is either a clip-like piece, as shown in the tenth photo from the top in the link provided by nailbender, or it is an "L"-shaped piece of metal, as shown here:
http://www.candlepowerforums.com/vb/showthread.php?t=229604
You need a 5/64" Allen wrench to remove the switch from the Mag body. Pinch the rubber button cap and pull it off to reveal the bare switch. Stick the Allen wrench into the hole in the center of the switch and engage a set screw at the bottom of the hole. Unscrew about 3 turns. The switch should now be loose.
If the serial number of your Maglite begins with the letter "C", then you push the switch from the top and out the bottom (after removing the tailcap and batteries and pressing the switch to "ON" so that the little plastic switch post is latched in).
If your C Mag serial number doesn't begin with "C", then you have to remove a lock ring at the top of the Mag tube. Take the tip of a ballpoint pen, or use the 5/64" Allen wrench, engage one of the dimples in the lock ring, and unscrew the ring. Once the ring is removed, the switch can come out from the top of the Mag tube.
Check flashlightking.com for a replacement switch. It is probably easier than trying to make a new metal contact, unless it is the L-shaped piece.