There is a way to unlock an iPhone running iOS that has a broken home button and no access to Assistive Touch. Try the following:
# On your Lock Screen, swipe right.
# You should see the Search field at the top.
# Type Settings in the Search field.
# Your 'top hit' should be the Settings app. Tap on this.
# You should now get the Touch ID or Enter Passcode screen.
# Enter your passcode.
# You should now see your settings.
# Go to General -> Accessibility -> Assistive Touch and turn it on.
# Also, while your still in the Accessibility settings, scroll to and tap the Home Button setting
# Change the Click Speed and/or Rest Finger to Open settings to see if that also helps.
Good luck!