Cannot setup FaceTime

I have tried to set up FaceTime on my iPhone5S running iOS8.1

I can call my wife on her iPhone4, but she cannot call me. We are both on the same WiFi network at home.

One guide I consulted said Settings > FaceTime "On this screen you’ll see any phone numbers and email addresses that can be used to contact you over FaceTime."

On my wife's phone this shows her mobile number, and an email address. On my iPhone there is a button "Use your Apple ID for FaceTime" but this fails with "FaceTime Activation Could not sign in. ⋯" The iPhone is connected to iCloud using the same Apple ID.

(I do have another Apple ID, used for Family Sharing., which I don't want to use, but this fails too.)