I had the exact same problem, and posted up some notes in
another subsim thread here.
My error was to confuse *online* (ubi.com) accounts when I did the re-install. First I created a *new* account (total mistake), and then slightly bizarrely used what I thought was the correct online ubi.com account but had in fact unwittingly created two of those when I'd installed the game in thefirst place.
The key fact (pun intended) is you should *NOT* be prompted for your CD-KEY on a re-install.
When the Ubisoft Games Launcher prompts for your online account and password, if the password is wrong (or account doesn't exist) then the Launcher will tell you immediately so I assume you are passing this stage.
With a correctly authenticated Ubi.com account, the Launcher will check the *online account* has a valid SH5 licence (from a CD-key typed in when you did the first install) and will proceed with the SH5 launch without prompting for the key.
So this points to the Ubisoft account you are using does not have a SH5 key registered. The only people that can look this up are Ubisoft support - give them your online account name and password, and they will ask you to confirm some other personal details. They will then confirm that account doesn't have SH5 registered (probably). At that point you can tell them the CD-KEY and they can do a search and see which online Ubi.com account *does* have that key registered, and with a bit of luck this is still an account you have access to. In my case they were able to send a reset-password to the email address associated with that account and I could still access that.
Let us know how it goes.
Bamzors
p.s. I'm guessing from this that all SH5 DVD's are actually identical - it's the cd-key printed on the back of the manual that's the only unique thing.