I would like to simply show a border around the currently selected item in my LongListSelector. I have set an ItemTemplate for my LongListSelector, but I am unsure of how to
When you access the selected item, you should access it as a border and not as a LongListSelector because that's how you show each item, while the LongListSelector is the container. You also forgot a semi-colon on the 3rd row, I've added it for you.
Your new code would be:
private void recent_SelectionChanged(object sender, SelectionChangedEventArgs e)
{            
    var item = sender as Border;
    item.BorderBrush = App.Current
                          .Resources["PhoneAccentBrush"] as SolidColorBrush;
}