I've fixed this issue on a couple of cars in different ways...
First off is the white box (immo) Did you ever try unplugging the immobiliser box's plug & plugging it back in again?
The contactors on the two halves of the plug are made from different metals, which react together in the presence of oxygen - a galvanic cell.
I fixed mine by unplugging and plugging in several times, then spraying with electrical contact spray to prevent corrosion. Was totally reliable for 4 years after that.
Now the reader coil. Just so you know, the reader coil is not coded to anything. As far as I understand, they are purely used to energise the key's transponder through induction, and the transponder then transmits to the immobiliser box, the immo box then "talks" to the ECU.
I found the problem was in the IDC connector that connects to the reader coil around the barrel. IDC = insulation displacement connector - this is when the crimp contact connects to the wire by cutting through the wire insulation. I.E. it's a mechanical connection, not soldered. The wire to this connector was very taught which had caused a bad connection at the crimp. As it was constantly being pulled. I soldered each crimp to each wire (2 in total) to make a good connection. Also pulled some of the immobiliser wire back through towards the coil and cable tied this to the barrel to try and stop it becoming tight again. Think this happens over years of steering wheel adjustment.
