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The option to stream games on your Apple TV using the Rogers NHL LIVE app should be available. We definitely want your experience with the app to be as seamless and stress-free as possible! Are you able to get past the login section on your TV? Do you receive any specific error message?
You mentioned that you are able to mirror the games from your phone/tablet to your Apple TV. Are you also able to login to your account and stream games via a web browser at rogers.com/nhl without any issue? Please give this a try and let us know.
I noticed that another user @newestguy also had the exact same issue and provided some feedback on how they were able to get it resolved. Please have a look HERE to see if it works for you as well!