= I have had this problem =
With the iBook I saw, it started at full bent-back then got gradually worse and worse. Eventually we tried to fix the inverter cable. Sadly that repair failed and the computer hasn't been used since. (a point you might want to bear in mind!)
With that iBook, when you shine a light at the screen, the image could still be seen underneath as it was only the backlight that had turned off. I would recommend following [http://www.ifixit.com/Guide/Repair/Installing-iBook-G3-12-Inch-Inverter-AirPort-Cables/1521/1|this guide] in order to make sure the inverter cable is plugged into the motherboard properly. If this doesn't work, you may well need a [http://www.ifixit.com/iBook-Parts/iBook-G3-12-Inch-Display-Assembly-Opaque-white/IF150-062|new display.]