Best course of action that I can think of would be to take a look at eBay and see if you can get the part from there and then switch it out with the broken part on your machine.
Most laptops come apart easily enough with a set of small screwdrivers. Just take your time and make a little map that shows where the each screw came out of so you can put it back into the correct place later.
It would also be a good idea to backup your data before you begin this process. As the laptop display is not working, you will need to connect an external monitor to do this. External monitor will also make a nice work-around so you can still use the machine while you are waiting for parts to arrive.