Get your new SSD in a USB enclosure and clone your other laptop running 10.5.8. Now swap this onto the failed macmini. Carefully replace the old HDD into the USB enclosure - and it will appear as external storage.
ideally, you could USB-boot with some linux based rescueCD (http://www.system-rescue-cd.org/) - use the software called gparted. This can clone your existing HDD into SSD. Just a try... your old hdd may give up or just allow you to copy. see https://gparted.org/
you can use the so called (trial version is OK) Carbon Copy Cloner https://bombich.com/