Is there a way to recover data from un-allocated space? I can send any kind of log if required.
Any kind of disk\partition manipulations should begin with data backup,
but I think you've figured that by now.
In order to recover any data you should first stop using this HDD (don't
mount it as writable) and get yourself another HDD with suitable size,
that will serve as destination for recovered data.
You'll have to remove HDD from your notebook and connect it to a PC with
standard SATA cable along with destination HDD.
With that setup, you have to use R-Studio software (They have free
version for Linux partitions, IFAIK) to scan source HDD for traces of
partition table and LBA offset of partition that was "sda8".
If R-Studio will manage to find right offset and recognize filesystem
then you will be able to start automatic scan for files available for
full or partial recovery.
Success is solely depends on how destructive was HP system recovery process.
Depending on how valuable your data was you probably will have to resort
to professional data recovery services, which will do roughly the same
procedures as I described above and charge substantial amounts of money
for it.
Since you won't write anything to your source HDD and it is not
mechanically failing, it is safe to try to recover data by yourself first.