I think the best option, at this point, is to consider the data rather than the operating system.
The easiest option would be to purchase an External Enclosure
and install the HDD from the broken machine into that. It's easy to accomplish and will allow you to access your files as if they were on a USB flash drive. Copy your important files to a good machine so they are safe, then you can consider either trying for a System Recovery from the recovery partition, or ordering the recovery media from HP/Compaq and doing the System Recovery using that.
While using the recovery partition method does offer the option for a "non-destructive recovery" which is supposed to leave your data intact, this is not always the case and it may be damaged or deleted.
A System Recovery using either method will likely take the machine back to the day it was purchased...any programs and updates that were installed after that will have to be reinstalled.Information on System RecoveryObtaining Recovery Media