If you've got $40 or $50 you can buy a keyboard with a connector just like your iphone charge cable (lighting connector I think they call it). I bought this one: http://www.amazon.com/Keyboard-Lightning-Connector-Retina-connector/dp/B00IQEL9N O/ref=sr_1_1?ie=UTF8&qid=1429235055&sr=8…
My phone screen was SMASHED! Home button didn't work, screen and digitizer broke, bad shape. But I used the keyboard to blindly enable iCloud backup, sign up for a 20GB backup plan ($0.99 per month currently), and then fully backup my iphone. Have patience, but it works and seems to replace it.
1. Plug in your keyboard to your iphone where you typically plug in your charging port.
2. Hold down your home button until Siri beeps to life (or hold the top left button on the keyboard that looks like a square) and just say "Voiceover On". Siri should respond and tell you siri is now on.
3. Press the "Lock" button on the top right corner of the keyboard. Siri says "pass code failed" as soon as I do this indicating I can now put in my passcode.
4. Type in your 4 digit unlock code on the keyboard (no need to press enter) and you should hear siri's voice over start talking about whatever app or text is highlighted on your phone.
5. Press the left and right arrow keys on the keyboard to navigate the various iphone apps. The iphone speaks whatever icon or text it's on. For example, you can scroll until you hear siri say "settings". If you have or can plug into another users iphone, this helps tremendously to visiualize what's going on.
6. Once you get the app you want press and hold down "Control" and "Option" keys and then press space bar. This is the equivalent of "clicking" on whatever item is highlighted.
Follow online instructions to enable icloud, backup to existing storage, etc.
Hope this works for you! I was very relieved to get mine backed up since I screwed up and didn't back up early.