Perhaps it has something to do with the connection from the batteries to the device? I would make sure the battery holding area is clear of any debris and try again. That, or you could look into the power button itself. The connection on the power button might not be getting a good enough connection to turn the device on.
If the device is completely unresponsive, utilize the reset button located on the back of the tablet. You will need something very small such as an unfolded paper clip or a pin/needle to depress the reset button. Coby Kyros MID7012 Troubleshooting