Remember you will also need the actual key part changed if you want to use it to manually open the doors or put in the ignition barrel. Shouldn't be too expensive to get a key cut though.
For that one i paid £110 for a new button and blade but i have another button bought from ebay with someone elses key,, im just using it as a spare and as far as the key blade goes, i have it with a standard key,, i have an OEM (clifford) alarm which uses the remote so thats why i need the extra buttons.
What you have done is used the remote (which can be programmed to any vehicle) not the immobiliser (which is vehicle specific).
So as the experts have said its not possible to buy a used key and have it fully functioning on another car. It is possible to buy a used key, remove the transponder and blade from a working non remote type key, have it fitted into the used key and then programme the remote into the car.