Do you still have your old computer, where Lightroom was installed? You can find it there.
I tried to re-install it but nothing happens and I can't get a serial number from it unless it re-installs can I?
You should be able (If you still have the old version installed) to open That version
go to Help > System Info > This should show your Serial Number.
Hope this helps. If not and you still have the proof of purchase, you should contact Support
for assistance in activating the Upgrade.
Hope this helps.
If you have not specifically deleted the info on you old computer it is probably still there. See the link below for where they would be located on your system.