Creating a backup from an iPhone with a broken screen which has also been shut off

Hello there.

I really wish to make a backup from my iPhone SE. The phone's touch screen has stopped working (you can see the locked screen, but it no longer responds to touching).

It also ran out of battery and therefore I need to punch in the code to reopen it - which is obviously not possible hard when the screen doesn't respond. When I plug the phone into the computer, a message pops up on the phone saying I should punch in the code before I can use any external deviced with the phone.

The start button works, though.

For some reason the phone also no longer connects to my wi-fi network at home, so I figure I my only hope is to try to back it up via iTunes. But the phone does not register in iTunes - which by the way is a newly downloaded version of iTunes on a new computer (the old one had a serios malfunction and had to be scrapped).

Any recommendations would be highly appreciated.