First off,

I AM NOT RESPONSIBLE FOR ANY DAMAGE TO YOU, YOUR IPOD MINI, YOUR MENTAL STABILITY, ETC. THIS TUTORIAL IS NOT ENDORSED BY APPLE, AND FOLLOWING THESE INSTRUCTIONS WILL LIKELY VOID YOUR WARRANTY.

Now that I've gotten that out of the way, here's what you need to do. It's actually quite simple.

Step 1. Switch your iPod Mini to 'hold'.

Step 2. Follow one of the hundreds of guides online to take apart your iPod Mini. This is a good one.

Step 3. Disconnect the battery for easy access to your cracked/broken/perfectly fine LCD screen.

Step 4. The screen is attached by four tabs. Pull each tab out. This will require a bit of work. If your old screen is already damaged, there's not much to worry about. Just make sure you don't break the circuit board!

Step 5. This is the tricky part. The LCD screen is attached to the circuit board by a ribbon cable. The cable connects to a brown rectangular piece. Carefully lift up the light beige section of this brown piece. It will pop up. Pull out the old ribbon cable

Step 6. Feed in the new cable. It won't slide in all the way. Snap the beige colored piece back down, and snap the four tabs on the sides of the screen back onto the circuit board.

Step 7. Put your iPod Mini back together according to the directions on whichever 'taking apart your iPod Mini' guide you followed.

Step 8. You're done!

If you need a new LCD screen, search on eBay. I found one for only $48.00. Finally, if you have any questions, feel free to ask! :-)