If you can't fix the home button, you can do without it quite well.
It's on by default, but make sure Settings > General > Multitasking Gestures is turned on.
Pinch inward with four or five fingers to get to the home screen.
Pull upward with four or five fingers to open the app switcher (multitasking).
To get a virtual Home button, and the ability to take screenshots:
1. Go to Settings > General > Accessibility
2. Turn on Assistive Touch
3. A bubble will appear, tap that to reveal hardware functions such as the home button.
4. To take screenshots, navigate to Device > More > Screenshot.
Hope this helps!