Unfortunately, there has never been a download for PSE5. By that time, the license number was not kept on an activation server by Adobe; it was only linked with your physical CD.
On the other hand, even if I have been able to test reinstalling PSE5 on my Win 10 PC, there are known compatibility issues.
It's time to upgrade to the recent version which is PSE2020. You are entitled to get the 'upgrade' price (- 20%) but you might even find better deals from big Web resellers.
Another question is how to move your photo files and catalog(s) to the new computer. Do you have backups?
Even if your old computer has died, the hard drive is probably still valid?
It's possible to convert PSE5 catalogs to the current format, but that requires additional steps. Ask if you need to.