At this point, I think you need to confirm that there is a valid update for your iPod, because as a rule, tapping the download button should complete the update automatically. It shouldn't fail.
Menus differ, according to the model of iPod, but begin by going into the Settings menu and and look for the heading General. In there should be About and look for the version number, or Software Version. On the 6th gen Touch for example, it's in Settings/General/About>Software Version and that states 12.4.3. (The heading below the software version gives the Model Name as well.)
The 5th generation iPod Touch can only install up to iOS 9.3.5.
Which version of iOS is currently on the iPod? That will give a clue about which model of iPod you have.