Press home to upgrade but home button is broken

There is nothing that you yourself can do to fix it.

Make an appointment at the Apple Store or take the phone to an authorized Apple Service shop to let them examine the phone and go over your options with you.

You can find a list of authorized service shops in your area using the link just below:

Make an appointment at the Genius Bar to have it fixed.

Genius Bar Reservation and Apple Support Options - Apple