From the description of your symptoms it sounds like the LCD itself is the failure point. In between the layers of glass, masks, liquid-crystals and more glass is a grid of transparent conductors. three for every row, RGB, or Red Green and Blue the component colors of light. On the opposite side of the liquid crystals is another grid, but with only one conductor for all three, this is known as the common line. The displays circuitry places a voltage across the front three changing the color of the pixel and the minute current is returned via the common. Now if the common line fails, an entire column, or row of pixels fails in the full open position, giving you that white line.
The iPod itself has no idea this is happening, it is just commanding the display and listening to the touch screen. This is why you can still control it, but can not see the interface. If you can connect a set of AV cables and get un-corrupted video you have even more proof that the display is at fault.
Consider replacing the display using the guide found right here on the iFixit site [guide|3307]