No need to open the jack. The spring clip should be visible on the inside edge if nothing is concealing it. Otherwise you'll have to remove whatever is hiding it first - maybe the motherboard has to come out.
Before going that far, try plugging your headphones in and out several time in succession to see if it that fixes the open circuit.
You already tried re-installing sound driver I assume? Could in fact just be a driver issue - - coincidence with headphone usage may be just that, a coincidence, when it's actually the driver.