How to Fix iPhone 4S Wi-Fi Grayed Out – With a Roofing Nail and a Blow Torch

I'd guess that the problem is solder joints under (in?) the chip cracking. The solder joints attach the chip to the circuit board, and also provide the electrical connectivity to the chip. A cracked joint means the electricity does not flow and so things don't work.

The "in the fridge" temporary fix is causing thermal expansion/contraction, which is providing enough mechanical movement to close gaps in the solder joints, thus getting the electricity flowing again, and making the WiFi work. The fix stops working when everything gets back to room temperature, meaning the cracks in the joint open up again.

The permanent fix is heating the chip until the solder joints melt, so the solder can flow back together, repairing the cracks in the joints. Hence the use of a bit of solder on the nail to monitor the temperature, so one can stop when it reaches the melting point of solder.

It's worth noting that such extended duration heating would be a no-no in production, where high yields are important, but in a case like this, do whatever works.