Probably the adapter is damaged on the A/C side, this is electrically independent from the USB side. Impact or force can lift the connector off of the PCB.
If the battery is dead the Cobalt 1 won't restart just from a USB connection. But probably you can restart it by connecting USB to a computer and updating the firmware using the instructions for the Cobalt 1 on the Atomic website. Another alternative is to completely remove and reset the battery. Either path should get the Cobalt to restart.
If you have the battery out and it's dead, then try charging the Cobalt with USB. This will charge the battery fine, but won't generate a reset pulse to restart the Cobalt. For that you need to then remove and reset the battery. It should restart when the charged battery is reconnected.
I think adapters (dongle) are still available from Atomic, and replacing the adapter is the best solution.
Let me know if you have any questions.
Ron