Have you tried restoring the device? It could be a software issue messing with your home button.
Or is the button actually stuck?
And what do you mean the device is locked for millions+ minutes? did you lock yourself out by trying the wrong password too many times? If so, you need to plug it into the computer its registered with to unlock it. But if its locked because the home button doesn't work, you can still use the top power button to turn it on... I just wouldn't go on any applications because, well, you won't be able to get back to the homescreen...