How did you buy the Pro purchase. Was it from the windows Store. If so was it an upgrade from Home to Pro edition. This type of licence does not requires a product key, it is a digital licence using you Microsoft account and uses your account as the key.
If you haven't looked already are there instructions in your purchase history to upgrade. If not then use the generic Pro key to upgrade form Home to Pro edition.
Make sure the system is offline. Go to Change product key in settings and enter the generic Pro key below:
VK7JG-NPHTM-C97JM-9MPGT-3V66T
follow any onscreen instructions then restart and go back online. Windows should activate using the licence you have purchased.