Does it work with the charger/power supply plugged in?
If no, then take the battery out and plug in the charger.
联想 Legion Y520-15IKBN 电池更换
If it now works with charger plugged in then replace the battery.
The charger should be 20VDC. See service manual page 21.
A defective battery could short out the system and prevent it from booting even with charger pluggged in.
A dead cmos/BIOS battery could also cause problems with booting. In the battery repair guide photo look to the right of the fans and above the hard drive. It is the disk covered in black plastic with black & red wire. Unplug the tiny white connector. It should be 3VDC.
Here is link to service manual:
Let us know how you go.