Since you've already extracted the broken part, it would be best to replace it.
I see 2 handy options:
1) Superglue and soda
This is pretty simple - you pour some soda on damaged place, add a drop of super-glue above. It instantly goes pretty deep to the soda. After it's solidified, you file off excesses.
Material is pretty sturdy (good enough for fixing plastic stuff), can take any form and can be done with household stuff (if you have superglue, which is pretty handy in itself). If you are fashioning replacement for the broken part (instead of mending it), you could also reinforce it with a loop of wire, but that is more of a trouble.
With a pair of pliers and some wire you can fashion a replacement. On the upside, it would probably last more than plastic analog. However, that just means that wear would be transferred to its socket, which is plastic just as well (and several years down the road you'd have a glaring hole in that place). So if you go down that route I'd recommend adding some sort of soft washer - plastic, thin rubber, even cardboard or several layers of paper would do.
Due to the same reason - wear transfer - while you could use soldering iron to apply wire patch to the broken loop (if the material melts, not burns), that wouldn't last.
Get yourself some wire roughly around 1mm. (remains of a sparkler, or coil from used notebook or whatever) Thinner or thicker would do just fine, as long as you're able to work with it.
Take a thin pencil, coil the wire tightly around it several times (coil should be as high as the loop, so 2-3mm), leaving several loose ends of couple cm at least. Pull the pencil out.
Now, imagine you're tying a simple knot. Put one loose end of wire trough the loop. If you can, maybe do the same with another too. This is supposed to hold different turns of the coil together. Part of coil that is wrapped around is the part that will be facing outside.
Then, squeeze the coil with pliers. This will make it thin enough to pass through the hole in keychain's body. If you stick a match / toothpick / q-tip middle in the coil, you can make sure that you don't accidently squish it flat. Or simply use the corner of your pliers.
Now you just have to spend some more wire on the part that would anchor the ex-coil inside the keychain, and don't forget some sort of soft washer.
If you find your wire breaking after you bend and un-bend it in the same place several times - heat it red in lighter's flame, this sort of resets the metal.