Just a word of caution.
On a number of occasions at work we have had customers try to do this type of repair and mix up the screws, if you are not careful you can push a longer screw through an important piece of the laptop just by tightening it up. Modern laptops usually only have one or two sizes, but older ones can have half a dozen different lengths. Also it can just be hard to remember where they go if it takes a long time to repair the power problem. I usually just draw little boxes on a piece of paper and write down where they go or draw a little diagram. Set the screws right on your notes. That way you have something to jog your memory when it comes time to reassemble.
When you have the board out, clean up the excess solder from your repair attempt. Then flip the board over. Look to see if one of the pins is sitting in the hole of a little donut of melted solder. I see this from time to time, I THINK it is caused by overheating due to a bad connection. If it is like this it will spark if you wiggle it powered (I try not to do that). Remove power and resolder the joint.
If you still don't get power across the board (anywhere but the jack), it is probably a mechanical failure of the jack. I usually test with a voltmeter from the fuse to the ground shield so I am not touching across the pins with the meter probes.
If it is a bad jack, apply ChipQik (it is stuff that lowers the melting temperature of the solder) and then use desoldering braid to remove the solder from all the holes to free the pins. I find the special alloy (chipqik) makes this job much easier. Because of the ground planes and the jacks shield, it is very hard to get to the normal melting temperature of the solder without damaging the copper PCB pads. ChipQik is your friend.
When they made your motherboard at the factory, they soldered it together by essentially sloshing molten lead over the bottom (in a very controlled way). It was not made to be fixed, it was made to be inexpensive to manufacture. You are the first person to probably ever touch that jack with a soldering iron, don't get frustrated.
Don't force the pins out, or wiggle the jack to get the pins out. You don't want to pull the hole plating out with the pin of the old jack. Seen people do that too! Be patient and take a break if it is hard to get the solder out. Sometimes, actually resoldering the pin and then trying with braid again will help wick out the hole.
Soldering in a new jack is much much easier than removing the old one, just use flux and try to be neat about it. Small chisel tip would be better than the large one in your photo.
If you succeed you saved your laptop, if not you probably would have had to spend 75 to a hundred bucks to have somebody like me do it, so it might not have been worth fixing anyway. It looks old just from the dust. I would definitely clean up the fan(s) and heat sink with canned air before reassembly if you get it working.
Hope that helps a little, I do them all the time at work. The first few were hard, but it got easier with practice. Slow down and be careful. Good Luck with your repair!