Hi everyone. I had similar situation with my iphone 5 but I was able to solve it. It stopped ringing or vibrating. It just would not do anything at all - screen would not flash, would not vibrate, it would just be quite. No notification at all on incoming calls. I tried it by calling my own number from other phones, it would just not work. I was frastrating like crazy.
Anyways, what I did is I did the software reinstallation:
0.BACK UP YOUR IPHONE FIRST!!!
1. Shut down my iphone (power off) by holding down the power button and slide the red power off slider on the screen;
2. With the iTunes application open on my PC, connect the iphone cable to the PC;
3. While holding down the home button on my iPhone connect the lightning cable to the iPhone and do not release the home button until the iPhone screen shows "Connect to iTunes" screen;
4. On your iTunes you will get a message that says that your iphone is on restore mode;
5. Click the Restore iPhone button on the iTunes. This will take approximately 15-20 minutes depending on your internet speed. Once the restore is finished the iphone will restart once or twice. It should restore your iPhone to the previous backup or whatever it had before. Or if not, you can resotre the back up you created before the iPhone software restore process.
Please let me know if this helps.