There should be a way to install again...what if the computer had a failed hard drive? The software should not be a one-time install.
Try this software to find the key on the first computer: http://www.gtopala.com/siw-download.html
This free program is called System Information for Windows, and will tell you quite a lot about the software and hardware installed in the machine. The SIW standalone version is what I use.
Once you start the program, you'll see a long column of options on the left side. Click on Licenses. This should show the licenses of various software installed on the computer. No guarantee that it will show this particular Cyberlink program, though on mine, I can see the key for PowerDVD using SIW.