I got this laptop a few months ago and ran into the same problem. After testing the end of the power cord with a voltmeter and getting the expected 19V bias, I figured the problem was inside of the computer. So I opened it up, which is not a very easy process, but is well documented on youtube by other users looking to install mSATA drives.
http://www.youtube.com/watch?v=Y7Ut1FGFj0sAfter pulling out the motherboard and inspecting it with the voltmeter, it seems that the problem is not the jack itself but the connection to the board; the solder was gone and the connector was waving around in the metal ring, hence the fact that wiggling will sometimes establish contact. Ended up having the re-solder the connection on the back of the motherboard. I put a big glob, so hopefully it will last.
This is a quite involved fix, but faster and way cheaper than asking ASUS to do it out of warranty. It's easy to get mad at ASUS for the problem, but this is quite common on many of these round connectors; the consistent wiggling and twisted will make the jack knock off the solder as opposed to a well-secured asymmetrical solution. Faulty design really.