There have been some issues when you try to activate an Alternate key but it stays in the state of failed, this might be because the internal process has created the index for the key but it didn’t stay as active in the UI and in the table that saves that information.
If you are OnPrem, what you will need to do is to go to the table of that entity and then on the index folder delete the index that was created. Then you go to CRM and just need to re-activate the alternate key.
If you are Online you will need to submit a Service Request, saying that the Alternate Key isn’t being activated and you want that the support engineer have a look into it.
Hope this helps