MY IPHONE 5 IS REFUSING TO RESTORE ITS FIRMWARE... WHAT SHOULD I DO

Well, to start off with, try describing what you have done, what you see on the screen, the exact text of any errors, including numbers, and what troubleshooting you have done.

What kind of restore are you performing?

When users specifically call out restoring firmware, there's a reason relating to a jailbreak...

Have you tried a dfu restore?

First: Put the device into DFU mode.

DFU MODE STEPS

1. Hold the Power Button (3 secs)
2. Continue holding the power button and also hold the home button (15 secs)
3. Release the power button while continuing to hold the home button (10 secs)
4. Your device should prompt with the "Connect to iTunes Screen"

Second: Plug it into your computer and open iTunes. It will either prompt to restore and up date or you can manually restore using the following steps.IOS Restore StepsBefore restoring:

• Verify that you are using the latest version of iTunes.
• Back up your device.
• Transfer and sync content to your computer.

Restoring your iOS device

1. Connect your device to your computer.
2. Select your iPhone, iPad, or iPod touch when it appears in iTunes. Select the Summary tab, and click the Restore button.
3. Click Restore.
4. After a restore, the iOS device restarts. You should then see "Slide to set up". Follow the steps in the iOS Setup Assistant.
5. If needed, restore your device from a previous backup.