Have you looked at the help on the HP website? I answered a question very similar to this and found the answer on HP's website. If you have the specific model number of the laptop I would be willing to take a look for you.
Also, is there anything notable that happened when this problem first occurred? i.e noises, sparks, any visable damage?
Did the laptop work with just the power cord and no battery before? It could be that the battery is dead and the laptop will not power up without a battery attached, even with power connected.
Ok, after looking on the HP site, I found a few things.
Firstly, it looks like they have issued a recall on the batteries for your notebook. It says that this problem should not effect the working of the notebook though, which is fair enough.
If my battery pack has been identified as eligible for replacement, and I have had no issues with it, could the battery pack have caused any damage to my notebook?
No. The issue identified is with the battery pack only and not with the notebook.
Follow the steps on this page (titled 'Getting Started') to find out if your specific battery is part of the recall. If you battery is part of the recall then you can get it replaced. While this may not be what is wrong with your laptop, it is worth doing to be sure (and hey, you get a new battery for free!).
Secondly, there is a long page helping you to troubleshoot problems with a non-booting PC. This link will take you halfway down the document to where it starts to troubleshoot problems with power. Follow the instructions from there and see if that helps you at all. Post back with any successes/failures.