Charge it with fast charger, preferably 2A charger atleast for few hours and turn it on while plugged to charger. Avoid fully draining batteries, it's bad for batteries, as well as cause similar issues.
If its turning on with fast charger plugged in and turns off when you unplug the charger, hopefully weak battery might need replacement, since its already very old model.
Otherwise, it could be power IC related issue.