What is the URL of your web application?
What did you set as the redirect URL of the authorization flow?
When is StartApplication called?
What HTTP requests are performed after the user successfully logs in (you can see it in the network tab of your browser's developer tool).
Usually not. Sphinx will automatically call the redirect URI you setup after the user is logged in. When calling that URL, if your code executes StartApplication, it should work.
But yes, you can also use OnUserLoggedIn event, that's what it is for, it's executed when the user is logged in, regardless if because it entered the user name and password, or simply because it was already logged in when the application is started (more specifically, when the TSphinxWebLogin component is instantiated and loaded from the data module.