Desoldering a tricky DC jack
The jack is faulty, so...
Cut all the plastic away with side cutters, leaving just the metal terminals. Now you can remove each terminal separately.
Hold the circuit board vertically in a vise. Get your iron hot as usual, and apply more solder to the pad to improve heat transfer. Grab a terminal (on its edge to reduce heat transfer) with your pliers, then heat the pad with your iron. The solder should melt quickly, then you simply pull the terminal out of the board.
Finally, use a solder sucker and/or desoldering braid to remove excess solder from the hole.
The pump is your friend. Incorrect technique with the pump is your problem:
The sucker has proven all but useless, the moment I move the iron out of the way to get the sucker in place, the solder has already rehardened.
Of course, you must apply the pump while the solder is still molten. Keep heating the solder with the tip of the iron, then bring in the pump, quickly press the button, and remove both tools.
Be sure to remove solder from the pump every couple of uses.
Solder pump tips are made to withstand some heat. You can do this without jamming the pump tip into the soldering iron tip: basically, just approach the tools to the solder blob from different angles. The pump creates a small area of vacuum around its tip; it doesn't have to be in contact with the solder blob when you trigger it.
Apply the suckage to each joint multiple times. Wicking is not going to be of much use here, because wick will not remove stubborn solder from the inside of a plated through-hole. Wicking is more useful for cleaning up solder pads when a part is removed, or fixing accidental solder bridges between closely spaced pins or traces.
If the pump has done all it can and the part is still stuck, there are ways to loosen it with the equipment you have. You can heat the pins on one side of the jack together at the same time while applying pressure to get them to move a little bit. (Or while pulling on the part from the component side.) Then do the same on the opposite side and repeat. By alternating this way, little by little, you will get the part out.
Well, repair of consumer electronics is off-topic, but the general problem is that there are a LOT of copper planes (including inside ones) in the area of the power jack of pretty much any computer, and they act as a very effective heat-sink.
You need to pre-heat the whole board, (in an oven) then heat the area you are trying to remove the jack from further. In most cases "removing the solder" is fruitless (you won't get it out where it counts, inside the board), you may well do better to add some solder to get better heat conduction - and then yank the jack out as fast as you can. Worry about removing solder after the thing is out, so you can get the new one in.
While a common enough problem, it's not that simple of a repair - it's not uncommon to damage the board in removing the jack. If the thing was well designed it wouldn't need replacing, so you know ahead of time that there are issues there.