Q: My ipod 5 won't turn on and my home button is broken so I can't restart it!!!! What do I do?

See my reply in this discussion

  Turn on without Home/Power Button

As I said, Click on the link in my previous reply and go to my reply in that link

Connect to iTunes,

Restore using iTunes

If you've synced your device with iTunes, you can restore your device.

1. Connect your device to the computer you synced with.
2. Open iTunes. If asked for a passcode, try another computer you've synced with, or use recovery mode.
3. Wait for iTunes to sync your device and make a backup. If it doesn't, learn what to do.
4. Once the sync and backup have finished,click Restore [your device].
   
5. When you reach the Set Up screen while restoring your iOS device, tap Restore from iTunes backup.
6. Select your device in iTunes. Look at the date and size of each backup and pick the most relevant one.

Get more help

If you can’t force restart or use recovery mode because of broken buttons, contact Apple Support.

BooMonster the poster can't use Recovery mode as you stated since that requires use of the Home button that the poster says does not work.

My link provided a means of placing the iPod in Recovery mode via an app on a computer.

My method doesn't involve recovery mode or a home button and uses no 3rd party apps. Mine uses iTunes without recovery mode.