I had this problem on older iPod Touch - that is why I bought iPod Touch 5th.
Anyway, if you want the button to get working back again, you have to take it to an Apple Store (there might be other repair shop in your country tho.)
One solution to this problem is to use AssistiveTouch.
1. Open "Settings"
2. Go to "General"
3. Go to "Accessibility"
4. Scroll all the way down until you find "INTERACTION" section and "AssistiveTouch."
5. Go to "AssistiveTouch" and turn it on.
Now you will see a black square with big white circle somewhere on your screen
which you can tap on it to show what it can do, or drag to put elsewhere.
This option enables you to click your home button via your screen.
You just have to touch the square and tap on "Home." Of course, double tapping on "Home" brings up App switcher.
The drawback is that when you play games this might be annoying.
If you don't mind, however, I think that is the best way to "click" your home button without taking it to the Apple Store.
Hope this helps.