It's so irritating when a sewing machine needle breaks! Sometimes I change the needle because it's getting blunt. However, I never throw them away, I put them in the drawer with the packs of new needles. Why? Well, if I need to hang picture (not a heavy one, mind you), a broken sewing machine needle does a great job. Just use pliers to nip off the end of the needle, so that you have the thick piece plus about a quarter inch of the thin part left.
Hammer into the wall. and there you are! Being made from steel, the needle should go into the wall quite easily.