Unlock broken iPhone with mouse?

So I managed to crack my screen, but for a while it was still working. Then however, my friend managed to accidentally shower it and now the touch screen is not responding (well, sporadically but not nearly enough to use the screen).

I have been planning to get a new phone for a while so it's not the end of the world, but I would however like to do a backup so I can get all my contacts and that.

After some Googling I have found that it isn't possible to backup my phone with iTunes without unlocking it and therefore I was wondering if it is possible to use a USB to lightning adapter to connect a mouse (you know those with a usb cord that you normally connect to a computer) to use the screen. I know it works with android phones and USB to mini-usb adapters, but does anyone know if it works with iOS too?

Thanks.