Turns out the cable did not break. It seems to have slipped at the adjuster, rendering it useless.

You can release the latch with a long screwdriver. You have to look through the grill with a flashlight from the left side (facing the grill). You can then reach in with the screwdriver, going thorugh the grill at about the 3rd or 4th row down and the 3rd or 4th column from the left. You then push the bottom of the latch and, viola, it releases.