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You might wanna try this:
  1. Hold down the Home button to activate Siri. Say “Turn on VoiceOver”.
  2. Press the Home button twice to bring up the passcode screen.
  3. Use the portion of your screen that responds to touch to swipe left and right, which will move your VoiceOver cursor to different passcode number buttons. VoiceOver will also say the button is selected, which is helpful if you can't see the screen.
  4. When the cursor is on the next passcode number you need, double tap to “tap” the selected passcode number.
Is this possible even after having shut down my phone? Holding the home screen button to activate Siri asks me for my password to enable her.
 
Update to anyone still here to help:

I have utilized Siri to allow me to get back in, but can now confirm half of the screen does not respond.

Where do I go from here?
 
Have you opened the phone and inspected the internals as already advised earlier in the thread?

If not do so, whether its yourself or Apple.
 
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Have you opened the phone and inspected the internals as already advised earlier in the thread?

If not do so, whether its yourself or Apple.
Unfortunately, I'm not nearly qualified enough to do so.

Do Apple charge to open the phone, or just to fix it?
 
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