Is the noise with the window shut or open - only ask that as you say the noise goes away when you put your hand over the mirror but then mention removing the doorcard suggesting it is when the window is shut.
If it whistles when the window is 'shut' - check that it is fully shut and is not staying partially down, as they do when closing the door. Otherwise, the only suggestion I can make is to check the rubber door/window seal, particularly around the window area, to see if that might be the problem.