Why doesn’t my Ericsson W518a not work?

This problem looks like a dead battery. Try replacing the battery. If it does not help, remove the battery and clean the contacts.

Because your question mentions that, when you try to turn it on, the screen gets a little brighter, then turns off again, it means this:

#1: Crashed firmware or EROM - full flash needed - bring your Walkman to service*

#2: Damaged LCD flexible cable - display replacement needed - bring your Walkman to service

• The service dealer can ask you for more fees, because this is old phone and all data will be deleted.

I don't know about the price of flashing firmware in USA, but in Slovakia the service dealers don't have a tools or hardware for these old phones.