If its a very cheap bike lock, why not just cut the lock open and throw it away?, your just gonna waste time trying to get something out of it, when your unsure yourself, and might not manage, They are selling bike locks for £1 in tescos (UK)
Dont pick locks that you need
I bought a bike lock just for picking
one question. . .has you tried putting the original key back in? see if the broken piece has gone right into the back. . .
dont push to hard incase you make it worse
if so you could still cut it off and pick it (i gather)