I have a different mobo, but I lost onboard sound occasionally too. Think I have it licked, but hasn't been enough time passed yet to be sure. ANyway...
What Intel told me to do (it's an Intel board in a new build), after I'd exhausted other possibilities, was to clear the BIOS and do a BIOS recovery. They said a damaged BIOS could cause the onboard sound to disappear. HP will probably have a different method than Intel of doing this, so check your manual, but the principle is probably the same. If you do go this route, just make sure you go into your BIOS and write down all your custom settings.
I'm bringing this up as a possibility, because you said this happened after "took it apart to clean it". If static electricity got to the BIOS while you were cleaning, it could be damaged. Try helpmeout's suggestions first--if they don't work, you might consider this.