Check the power adapter, or try to change another normal power adapter.
Check the battery, or try to change another normal battery.
Try to remove the battery, and only connect the power adapter. Or only use battery, and do not connect power adapter to test it. Then check the Power supply Circuit and switch on/Off circuit according the test result.
If the circuit has not any problem, and it still can not turn on, please try to discharge.
Open the shell of the terminal. Do it according to following picture.
Strongly recommending (only for version V1.1): Remove the button battery forever. But for this way, the device can not keep the date and time. You should change the firmware to get date & time from your web server.