Is it stuck in a loop where it keeps showing the Apple logo and then disappearing? Or, even worse, not coming on at all? This is very common, and very frustrating. But if you're patient and determined you can fix it. I've fixed a half dozen T5s over the last year with this same issue.
What you need to do is put your iPod into DFU mode, then try installing the firmware. It may take several attempts before it works, but it will eventually work. This website explains how to get into DFU mode: Plug your iPod into your computer with iTunes running and turn off the iPod. Hold down power for 3 seconds, then while holding power also hold down home button for exactly 15 seconds and release power but keep holding home. You won't see anything on the screen but iTunes will complain that it's found a device in recovery mode. In iTunes choose the device and restore it.
Protip: If you have a Mac you can open Console and monitor progress. At first you'll likely be getting errors and the iPod won't even report it's serial number. But gradually after a few attempts it'll get further along the process. As soon as the firmware fully re-installs it'll successfully reinstall iOS.