Researched same problem for HP dv9543/9500 9000 series;
Had similarly hoped it was the power jack feet with broken solder...unfortunately not. On dis-assembly I found that the socket is designed to preclude this by having its own mini-board and then a cable goes from there to the motherboard.
It appears to be likely a charging chip or similar.
You likely can run the laptop by charging the battery, which tells you that the laptop is otherwise functioning normally.
As a stop-gap solution you can get an external battery charger that will connect to the battery directly (EBay); two batteries and you can alternate.
And/or the HP-Dock which may power the laptop via the cardbus slot with a cable connecting them (as apposed to the direct docking); and the AC adapter you have, plugs into and powers the HP-Dock.
Also saw on ebay a vendor that would fix the charging circuit section (or other issues) for $75 I believe.
Unfortunately, my laptop was with the offspring overseas in the jungle, and had to send another motherboard to him via a friend.
Further unfortunately, the replacement motherboard had the graphics chip heat problem alluded to by the first answer person; and the graphics chip had to be used from the original motherboard.
The laptop failed within warranty but was of course not here nor was I told of the problem until after expiration; have 3 of the same model laptop--flagships of the time with two hard drives.
another ebay outfit will reinstall the NVidia GSM-8000 or something graphics "board"/chip with a copper pad or something to solve the heat dissipation problem.
(now it is back nearly a year later with the video off, so assume the chip overheating problem--had it working for awhile--now off)
Have to refind the vendor that solves the graphics heat dissapation problem or tear it down again and see if amenable to ArcticIce or similar--though probably doesnt just snap on and off..
HP dis-assembly manual, by the way, does not beyond reaching the MotherBoard...mostly for replacing major modules (screen/HDD/Keyboard/WiFi/speakers/etc), but does give the major Mother Board part numbers (Intel/AMD, w and w/o graphics "cards" and such)
hope the info may help