DC jack removal/desoldering - impossible

So my laptop has been playing up for a while now after my cat jumped on it and it fell onto the plugged in charger tip. I would charge but only when the charger was held a certain way/wiggled into the port.

The plastic in the DC jack was cracked so I ordered a new one and got a soldering iron.

Having disassembled the laptop and removed the motherboard, I have now spent hours trying to get the broken jack desoldered from the motherboard.

The two pairs of side pins holding it in eventually melted away after I took a new iron to the contacts.

The three 'top' pins holding the jack onto the motherboard seem impossible to desolder. Making things worse is that the black plastic of the broken jack is melting and making a mess so it's even harder to work with as I am only able to place the soldering iron on one side of the motherboard.

Any tips or tools I can be using to otherwise dislodge these three pins?

I have tried placing the soldering iron in the pins and gripping the jack whilst wiggling in an attempt to ease it out but nothing is working. Two of these pins seem very clogged and almost cemented in.

I am at breaking point. Thanks in advance for any help or suggestions.