I do not see a redirect_uri specified on the Twitch.init() which I believe, if I remember right, is required for the authentication method. This is how I call Twitch.init():
Twitch.init({ popup: true, redirect_uri: $.panel.config.redirectURI, clientId: $.panel.config.clientId },
Since you are trying to login, that would require verifying the identity of your application, which the redirect_uri provides.I could also be entirely wrong too 