Laptops using AC power adapter identification communicate through a 3rd wire in the DC power cord. The signal arrives at the motherboard through a protective circuit shielding the POST controller from electromagnetic pollution.
Insufficient protection or simply excessive signal pollution can either destroy the protective circuit or the input of the POST controller. In either case, any repair attempt requires the motherboard to be disassembled to access this circuitry.
A simple test to check if the above is the case is to test the AC power adapter with an identical laptop. If battery charging resumes at least the AC power adapter is working. Meaning the problem can either be the DC jack or the protective circuit on the motherboard.
If laptop motherboard replacement is not an option, it's possible to still charge a laptop battery using a standalone laptop battery charger.
For the DELL D610 Latitude there's another interesting detail. The protective circuitry is also connected to a pin of the docking station connector at the bottom of the D610. In case the DC jack is the problem - the D610 docking station is another way to connect the AC power adapter to the laptop.
Back to the laptop battery charging diagram.