Is the L1 button not working or the L2? If the L2 button is not working, you can pop it off the assembled controller with a Spudger. When you do that, you should see the rubber buttons that the L2 button pushes down on.
Realign the rubber button and rubber strip into the plastic channel and then put the hard L2 button back on (pinch the sides of the button gently and then insert them back into the assembled controller, they should snap in place).
After that, try the L2 button; you should hear a clicking sound.