How Do I Get into My iPhone

I broke my phone screen ( the LCD and Touch capabilities ) and do not know how to get in. I need to back it up so I can apply it to my new phone. I cannot enter a password VIA touching the screen to put it in. I was actually planning on backing it up manually that day. I have had iCloud back ups, but these are almost irrelevant seeing as how they are a year and a half old. I ran out of space on my Free iCloud storage. I was wondering if there was any way to install a Jailbreak or something manually, to use some kind of Keyboard app or something. I have tried backing it up on my computer using iTunes, but that requires me to trust the computer (it is a new one I hadn't trusted yet). If you need any more info, let me know. Thanks.