Arduino Micro just blinks, won't connect to PC
I have something similar happen a couple of times with some atmega328s (on arduino unos). I believe it was the bootloader that some how became corrupted and so no longer responded to the computer. The only way to fix this is to reflash the bootloader using an ISP programmer. Luckily the arduino can act as an ISP programmer so all you need is another working arduino.

See https://www.arduino.cc/en/Tutorial/ArduinoISP for how to do this on an uno. The same can be done with the micro/nano as they all use the same atmega328p chip its just a matter of wiring it up correctly.

If you cannot reflash the bootloader with this method then one of the chips is probably damaged and would need to be replaced. At this point it might be easier to replace the whole board.