Hi,
Unfortunately neither HP or Microsoft will have any record of your Product Key - but you can order a set of Recovery Discs from HP.
Using these discs to reinstall windows will not require you to enter a Product Key - activation will be automatic.
A guide on using Recovery Discs to reinstall Windows can be seen on the following link.
http://h10025.www1.hp.com/ewfrf/wc/document?cc=us&lc=en&docname=c01867418#N720
Regards,
DP-K