OS 10.5.8... is this a PowerPC Mac or Intel?
Download this zip file if you haven't already:
Inside that should be a few folders. One is named "10_3_r183_90".
Inside that folder is a flashplayer_10_3r183_90_mac_intel.dmg
Double click that and it will mount on your desktop, and open a window with the installer.
Right click the installer and choose "Show Package Contents"
Go to: Contents/Resources and double click the Adobe Flash Player.pkg file
The installer will open, ask for your Admin password, and it will install.
That's the install part.
Trouble is, a lot of websites won't work with Flash Player 10 anymore. -Bad news.
The good news is: because you have an Intel Mac, you can upgrade to OS 10.6 and you'll be able to run a LOT of stuff that 10.5 won't handle.
Apple has it for $19.99, and they ship free. http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard
Mike, thanks so much for your help. Fortunately I was able to figure out what was wrong. Flash Player was installed all along, but I had unclicked the "allow plug-ins" box. Now why Adobe did not give me more information while I was struggling with this, I do not know. It would have been helpful for them to let me know that Flash Player was installed. My computer said it was, but the little Adobe "check to see if you have Flash Player" box kept coming up empty. Therefore the confusion. I appreciate your advice and will save it in case of problems later on.