Before you try re-soldering the board, eliminate the possibility that; 1. the power supply connector, or 2. internal corrosion could be the problem. For the power supply try another one. For corrosion use a small squirt of WD40 on a q-tip and apply to the most exposed area of the connector and insert - remove several times with power off. If either of these fixes it, you're done.
If your soldering experience is limited (if not, please excuse the following) consider "dressing up" the existing connection points instead of replacing the power connector. Actually replacing connectors isn't hard, but it's tricky if you haven't done it. I would use a 25 watt iron with a freshly tinned micro point (1/16 in round) for each connection point for no more than 5 seconds per effort. Place the iron point on the pin where it connects to the MOBO. You want to end up with shiny solder on the pad and up the pin. Try this on any connector on some junk boards first and, when you're satisfied with your work, try the worst laptop first. You'll know you're doing it right when each connection gets shiny in less than 5 seconds. Don't forget to tin the tip for good thermal transfer before every joint and heat the pins, not the MOBO pads.
What do you think zepper? Does this sound reasonable to you?
Jim