Why won't my screen work?

Cassandra, great way of checking:-) Normally I would have said that it is your inverter or your backlight that has failed. Since you found out that it works but it stays dark. The only thing that points another direction is the fact that " I also tried connecting an exterior monitor but it doesn't recognize the monitor at all. " this would be a problem with the video card. The external monitor and the LCD use a separate pathway and the external monitor would not be affected by a bad inverter or backlight. There is not really a good way to check the inverter unless you have the right tools. I think I would replace the inverter with the cheapest one I could find, something like this or from here and replace it by either following this video or by downloading the service manual from here. The order of failure for this would be 1. inverter, 2. LCD CCFL, 3. GPU (motherboard). So for your cost this could be anywhere from $20 to $100's depending what you find. Hope this helps, good luck.