I have a non responsive touch screen. Can I recover my data?

I purchased my Nexus 4 during the first Wave. I received my phone a day before I left on a tour across the country. Mid way through the tour, the device slipped out of my pocket on to the floor. It was it's first fall, and it was a fatal one. There is a large crack across the screen, and the screen doesn't respond to any touch input. The display still works fine though. But the glass seems to be pressed to close as I can see the liquid crystal move a bit when I try to use it. I did not have the device in USB Mass Storage mode and I had yet to unlock the bootloader. After numerous calls to both google and LG...they claim they can do nothing for me in terms of recovering the data.

Is there any way to recover what I have on the phone at all? Better yet, is there anyway to repair the screen? I will even go as far as purchasing a new phone, removing the main board from my original N4 and placing it in the new one if it would work.

EDIT: Figured I should update and thank anyone who's responded. The utility mentioned bellow is amazing. However, unless the phone has USB debugging enabled, there isn't much hope. As far has I know the N4 stock cannot function as a USB host. So for now, my N4 sits idle on a self until I decide to replace the screen. It's just amazing...four small cracks towards the top, and the touch screen is simply dead. I have an HTC One S where over 70% of the display is completely shattered, and shards are missing as well...and it still functions perfectly. Thanks everyone for the insight...and remember to turn usb debugging on : (