I had repeated problems connecting Facebook to Instagram on my iPhone 5. I would click “OK” when it asks for permission, then it says: “Facebook Login Failed. Make sure you’ve allowed Instagram to use Facebook in iOS Settings -> Privacy -> Facebook.” The frustrating thing is, I DID already have it set to “Yes” in iOS settings, it still failed.
Anyways, I found the solution on this forum post: https://discussions.apple.com/thread/4648799?start=0&tstart=0
The short version of what to do is here:
1. Log in to Facebook on your computer
2. Search for Instagram (type “Instagram” into search)
3. Click the “App” Instagram (not the page)
4. You should see an option “Send to mobile”
On your iPhone you should now get a notification, and it should work!
Thanks to shookim for posting this solution on the Apple forum.