If I understand right, your power pack has a blue light that comes on when you plug it into the wall but then goes out when plugged into the laptop.
I would start with trying with the battery removed, then I would try a different power pack (borrowed preferably), see if that works. Otherwise, it sounds like a power jack issue (which I believe is a rather involved fix), could be something else though too. When you do plug in the power pack to the laptop, does the plug feel as though it isn't in all the way or loose or at all unusual?