I'll discount the connections or cables and sockets but the 3 major parts are the battery (natch), charger and the motherboard.
Since you obtained the Dell supplied battery the issue won't be the non-maker battery issue. However the laptop appears to be in pieces so all I can offer is that this is a matter of figuring out what part it is. Most folk won't replace the motherboard it seems.
I have a d620 that will power on with the cord, but will not charge the battery. I have bought a new battery since this started happening so I know it is not a bad battery(unless the replacement was bad as well.) I have the laptop taken apart right now, so please help me from here!