In the comments a few people agreed that you really cannot fix this. I agree, even if you still had the broken piece gluing it back on would involve purchasing a plastic glue that would cost as much as the diamond case, and the glued joint would be nowhere as strong as the original hinge.
But you do need options and here are my suggestions.
Sooner or later you will come across a diamond case. You might see CDs or DVDS at a garage sale for cheap. You might loose one of your CDS and find you have a spare, a friend may have on laying around. You might pick one up at the goodwill for a quarter.
Until then use a rubber band to keep the case intact and closed.