A lock is made up of driver pins, that push tumbler pins, that are pushed on to the key by tumbler springs. Very unusual for one to break, and if they do they generally won't stop the key from going in. Is there a possibility somethere foreign has been pushed into the lock.
In any case, pushing on the key will not clear the obstruction. You probably have to take it into a locksmith, and/or replace the ignition switch.