Retired Electronics Engineer.Seven years in Nanotechnology Design.Taught City and Guilds Level 2 Electronic Servicing.Lectured on soldering techniques to Brother Industries and other local companies. Keen amateur Musician.
I had a similar problem with a 3gs. I tried a couple of spare screens but could not get rid of the dead spot. A narrow vertical area corresponding with the letter I when the keyboard is on screen. I put the main board under my microscope and found small traces of what looked like solder splashes on two of the pins in the digitizer socket.
I had the same problem with the first 5th. gen I took apart. My comments on the spring went unanswered too. Came to my own conclusions. First the odd looking bit of black plastic (which usually flies out when you pry the click wheel assembly out). If you look at the main board you will see slot about 5mm long by about 1.5mm. Take the black bit in some forceps and you will find it will click back into this slot. There is a hole in the black bit which holds the tiny spring. If you look at the ipod case, preferably with everything removed), there is a small patch of silver plating on the colored anodized surface. My theory is that the spring is a grounding contact between the case and the center of the click wheel assembly. That spring is a little devil. If you drop it on your workbench, even on a white cloth, its so small - just blink and its gone! The only way is to open up carefully then pickup the spring with a piece of blue tack or Scotch tape - and leave it there until you need it. Most of my springs...
You haven't mentioned what happpens when you connnect to your computer. Trying to do something with a dead device on its own never yields much info. Your PC is your diagnostic device for looking into the heart of that thing. Does your PC "Ping" when you plug in the USB cable. You may have to wait for several minutes.Your USB could be struggling to charge up a dead or shorted battery. If anything at all appears in the bottom RH corner of your computer screen then you have a ray of hope. Try this - if you haven't done so already.
If you think the fourth generation is tough you should try the fifth. I once had the task of mounting tropical spiders web on bits of half inch square card. We put them in a machine we were designing to measure breaking strains of picocoulombs.Nothing compared to that nano!
The iPod has a liquid crystal display. A very thin capsule with a chemical compound which builds up the picture under an electrical charge. Likely is this capsule ruptured as a result of the fall. There will no longer be any picture or pattern. All you are seeing is the electroluminescent screen (backlight) which shines through the crystals in the display. This must be replaced as a complete unit. Relatively easy on the 4th gen nano. The contents of the display are regarded as toxic so take precautions if you see any of this inside. Mop up carefully with camera lens tissues. After plugging in the replacement give it a quick try before pushing it all back into place.
Just noticed your question on the video cable.I have a similar problem. On my nano the video cable plug (which is on the end of the cable and part of the print), is damaged by electrolysis.(water damage) One of the pieces of track is missing.I have been able to get hold of some conductive silver paint(ebay).Hoping to be able to paint this on with a toothpick then scratch out the required bits with an engraver made from a sewing needle.(I have a watchmakers pin chuck for this purpose) In your case,you could try carefully overlapping the cable like splicing a piece of recording tape. Cable is not too short so a small overlap should not be a problem.Use superglue for the overlap and align accurately under a high power magnifier.Not sure but you may have to rub away the layer of varnish from the cable - to expose the bare metal.Paint on a small stripe of silver across all the tracks.When dry, carefully scratch out the spaces between the tracks and lacquer over.This is only a theory but worth looking at.