So I think I got something for you to get the device open. @flannelist did a very good job finding a replacement connector, but to kinda get you running so you can get the file off your device you can use Siri to turn on Voice-Control on your device so you can use it hands-free.
You can use the home button to activate Siri, or use ***"Hey Siri"*** and ask her to ***"Turn on Voice Control"***. From there just say ***"Go Home"*** and your password screen should appear. Then Just say ***"Tap*** [Insert button here]***"*** for each button in the password.