1) *It's highly recommended to make a backup IMAGE to a secondary drive (USB or local) if you can do so. If you screw up anything you can RESTORE the backup image.
If you have a WD drive you can go to the Western Digital site and find the Free Acronis True Image program. For Seagate the same program is called "Seagate Discwizard". You can also use the Windows backup utility but I prefer the above programs. (if you use the above programs set the COMPRESSION level to the second highest).
Roughly estimate HALF to 2/3rds the compression of the current C-drive capacity but it varies a lot.
2) Still confused if Windows 8 was pre-installed or not.
If so, there should be a BACKUP IMAGE in a hidden partition. If you run this as said from bootup and the proper F-Key, then this will replace the current Windows 8 version by extracting the backup. It will end up the same as when you bought the computer.
(then it will apply Microsoft Updates, and you may need some drivers and programs... possibly delete some pre-installed crap like games)
I know you said you "lost" the disk/key but it's pretty vague and not clear if you built the computer with a disc or not.
3) There is a way to install Windows 10 for free using the Assistive option.
https://www.microsoft.com/en-ca/accessibility/windows10upgrade
4) Consider buying an SSD if you don't have one. At least 120GB. See pcpartpicker (your Country).
- that also solves the BACKUP issue because you can:
a) install the SSD (so you can test that it works), then
b) shut down and unhook any HDD's
c) Install W10 Assistive (or W8.1 if using the KEY method)
d) Finish install
e) Verify it works, update etc.
f) Hookup older W8 drive then FORMAT it, then
g) make a BACKUP IMAGE of the new W8/W10 in case something goes wrong
OTHER:
**ASSUME that anything you do will cause a corrupted drive and plan accordingly. For example, restoring a BACKUP IMAGE could fail. Then what? Or the KEY you used may not work. Then what?
I can't stress enough to have either a:
a) backup IMAGE to secondary HDD, or
b) buy a new HDD/SSD so you can unhook the current W8 drive
One of the CHEAPEST solutions is a 1TB WD Blue drive, but an SSD can really help make the PC snappier. Unless you need the physical space then I'd get a 120GB or 250GB SSD.
Example:
120GB ($50USD)
http://pcpartpicker.com/product/qf98TW/a-data-internal-hard-drive-asp550ss3120gmc
250GB ($75USD)
http://pcpartpicker.com/product/gf98TW/a-data-internal-hard-drive-asp550ss3240gmc
As you can see the 250GB has 2x the capacity but only 1.5x the cost (roughly an extra $25 + tax).
And again, if you only have one HDD now this allows you to use the HDD as a backup. Always, always assume that your Windows drive can die at any time and have a plan for that.