that first line is confusing. did you mean an optical disk drive? "Although, the laptop does not have a DVD disk drive."
you moved the laptop hard drive to a pc and it didn't boot. that is normal, the machines are too different.
did you move your drive to your brothers laptop and it worked, or did you move brothers laptop drive to your desktop. swapping drives WILL NOT WORK on every machine, depends on the specs of the donor and the recipient, too many changes and windows freaks out during boot and well, hyphen.
I suggest replacing your drive back in your laptop before you damage the software therein. boot to a linux distribution from a USB, details below. boot to linux and mount the windows drive. download the windows installer for the game and place the installer on the desktop folder in the users folder.
once that is done shut down linux, remove the USB, boot to windows and install from the file you just placed on the desktop.
if you need to access a DVD-drive to install the game, there is always the network. joining both machines to a workgroup or homegroup. you can then share the optical drive on the desktop and it will be accessible through the network to the laptop. open explorer, network, computer name, and there should be the drive.
From ultrabooks to netbooks, computers are shedding their optical drives.
www.howtogeek.com
another option would be to make an ISOfile of the install disk, you can then use anything to move the ISO to the laptop and windows can mount the ISO as if it were an local DVD drive.
An ISO file is a complete disc image of a CD or DVD bundled into a single file.
www.howtogeek.com
Linux how-to
Boot to a USB drive with linux on it. grab a USB drive, a copy of rufus and a linux distribution.
http://distrowatch.com/ has tons of differing linux distributions and download links. I personally am fond of linux mint with cinnamon.
https://rufus.ie/ the utility used to extract the ISO file to the USB drive.
use rufus to extract the selected ISO to the thumb drive. it will make the drive bootable and you can run linux from the drive once done.
Reboot into linux and proceed to test the hardware. connect to internet, watch videos, await problems.
if linux is good and stable the issue is most likely inside windows or otherwise software related.
this is a test of the hardware.