I bought a new (used) display and digitizer from ifixit for an Ipod 5g. I had originally just purchased the replacement...

I had this same problem. There's a small black dot (very small, very easy to lose) that sticks up on the home button, and is what pushes down the button on the flex cable. I cut a small piece of VERY thin plastic and super-glued it to the top of the flex cable button, and it works like a champ. Like William said, though, you can use the assistive touch under settings/general/accessibility to turn on a virtual home button and place it wherever you want it on your screen.