Some possibilities:
1) Short circuit on motherboard -- If there is a short circuit, then the AC adapter will turn off when you plug it into the jack. Does the AC adapter have a light? If so, make sure it stays on when you plug it into the jack. You can also test the voltage of the adapter before plugging it in. Then test it again after unplugging it from the jack. Make sure it is still 19V (i.e. it didn't shut off to protect itself).
2) Bad AC adapter -- Even if you read 19V, the adapter can still fail under load. It would be helpful to try a different adapter or try this adapter on another Dell notebook. Just to be sure.
3) Bad solder joints -- You can remove the motherboard and inspect the solder joints. Wiggle the jack and watch if the joints move. Look carefully. Sometimes the whole solder joint (and pad) lifts off the board a tiny tiny distance. It's very hard to see. The pins may need to be resoldered. If you have soldering skills, you can resolder the jack just in case. If any pads are damaged, then that makes resoldering more difficult (the missing pads must be replaced with wires).
4) Bad jack -- There might be something wrong inside the DC jack. If so, it will have to be replaced.