All you need is Rubbing Alcohol.
Add a few drops of Rubbing Alcohol to the broken powder and mix it around. It should form a paste. I put a little too much Rubbing Alcohol in mine so it was more of a brown liquid than a paste.
Let the compact dry over night. The Rubbing Alcohol will completely dry and you should have just the powder left.
Ok so it doesn't look perfect, but its better than throwing it away :)