The 'Link a device' function can only be used once per village. After that, your village will be permanently connected to both your Game Center account (on iOS) and your Google Play account (on Android), allowing you to transfer a village using GC/G-Play in the future.
All you need to do, depending on your platform, is sign into the correct Game Center or Google Play account, then open the in-game settings and press 'Connect'. The game should now offer to load up your village.