Take it physically apart and check out the battery and the contacts of the battery to the circuit board.
You do know that the back comes off of these phones and that there is a replaceable battery right behind the back cover, right?
The battery itself may be dead or the contacts may just be dirty or somehow bent.
Also with the proper technique and equipment it may be safely possible to charge the battery OUTSIDE of the phone and then when it is somewhat charged you could put it into the phone to see if the phone boots and works.
It should also be possible to buy or borrow a new battery and try it in the phone.
Actually if you know the right technique you might even be able to power the phone through the battery's contacts with a suitable 3.3VDC well regulated power supply. The battery itself seems to be rated at 3.7VDC but you would not want to come close enough to that limit that you could accidentally go over it with a bad power supply.
I've had done that. The mainboard has many extra tiny SMD components which I guess may not be replaceable.
There are metal shields on every chip which look like they are soldered. I guess they are RF shields intended to minimize the radiation for decreasing SAR values, am I correct? You can see high-quality pictures at
https://globaldirectparts.com/oem-nokia-lumia-520-motherboard-at-t/How come there is no visible circuitry path on the board? is it sandwiched between the outer layers?
I don't have a multimeter yet but the battery seems to work as I connected an LED to it and it glowed.
One thing that I noticed earlier was that after powering the board, you can hear a very quiet low-pitched tone by bringing your ear next to one of the chips on the upper section of the board near the camera module.
Another thing which I noticed is that if I connect the board via a USB cable, there would be no voltage on the battery terminals of the board.
A friend recommended to check it with a power supply, mine (a cheapo adaptor) had only 4.5 volts so I did apply it to the battery terminals attached to the board after he said 4.5 volts would cause no harm. Nothing happened but now I'm a little doubtful if 4.5 volts would be harmful to the board's components or not.