Troubleshooting VGN-FW series VAIO Power Issues
Follow the instructions from the Audio card guide to get the system apart. Once you have the system apart, you want to unscrew the DC power connector from the board. Once you get it loose, it won't be able to come all the way out due to the magnet on the cable. For this quick fix, you won't have to though. If you look closely at the side of the connector that the wires come out of, you can see that there is a central wire to the spoke and a wire on the top that goes to a little V shaped piece of metal that is stuck in the top of the connector.
The problem that seems to cause intermittent power issues is that this V relies on holding it's shape to provide counter-pressure to the plug. Over time, the metal weakens and no longer holds a solid connection against the jack. I was able to fix it by simply inserting a flat head screwdriver to expand the gap. I was then able to push some aluminum foil in to the gap to reinforce the metal and give it a little more counter-pressure against future issues. Be sure that the aluminum foil won't fall out and cause a short when you put it in and I recommend testing the connection with a multimeter prior to reassembly.