My iPhone 5 won't charge or turn on

Thanks to my Apple lightning connector shorting out, my iPhone 5 has been unable to charge for the past 48 hours. Around 36 hours ago, I shut off my phone with 5% battery remaining, and let it sit while I tried to make time to run into an electronics store to purchase a replacement.

Today I purchased a third party cable to replace my old one, but it doest appear to be doing anything. I've tried charging though both the Apple wall adapter and my computer, but nothing has happened. My screen still remains black, and my phone unresponsive.

I'm unsure of what to do. Is there a hardware issue with my phone? Or, perhaps the third party usb cable is to blame?

I've tried holding down the power and home buttons for 30 seconds, and it's been on the charger for at least an hour. Am I missing anything?