Hi @unknownsheik
Most probably the battery has failed and that's why it won't charge or the phone turn on when the charger is connected. Try [https://www.manualslib.com/manual/163885/Sony-Ericsson-Z310a.html?page=11#manual|replacing the battery] and then fully charging it before attempting to turn on the phone.
Search online for ''Sony Ericsson BST-36 battery'' to find suppliers that suit you best. Searching online seems to indicate that this is the model number of the battery for the phone, but you can verify this, by looking at what it shows on the battery that is already in the phone.