Scan WhatsApp QR Code from device with broken screen to access WhatsApp Web
There is an application through which you can control the screen over USB and adb called ADB control. Follow the link to install and use it. Then you backup contacts to your phone/google account once you have control.

Source: http://marian.schedenig.name/2014/07/03/remote-control-your-android-phone-through-adb/