Is it the LCD that's messed up?

Try doing a hard reset by holding sleep and home for up to 10 seconds. If that doesn't bring it back, then it could simply be a loose LCD ribbon. Take off the back cover. Disconnect the battery which is held by one ore two screws depending on your iphone, then remove the very top EMI shield. Should be 4-5 screws. There should be two large ribbons to the right of the camera. Disconnect and reconnect. Reconnect battery and turn on the iPhone. If it still doesn't illuminate, disconnect the battery again and disconnect and reconnect other ribbons underneath the other EMI shields. If the LCD still fails to illuminate, then it's possible your LCD is damaged and will require replacement at that point. Best of luck.