Which app are you using to listen to "internet radio"? And why would you want the display to stay on during the broadcast, never mind after (it's radio, you are listening, not watching)? Why is the display going dark after it ends an issue (you can just wake it)?
In short, I think I don't understand your problem. But since there are many apps that can listen to internet radio, you should tell us which you are using (or try others and see whether they fix your problem).