Oh dear. How come it's working anyway?
I tried the phone without using the handset, just using the internal number pad, speaker and microphone, and that's all working.
The charging light for the handset is lit, but the handset doesn't work.
I examined the battery, knowing that the owner had changed it not too long ago.
It's an aftermarket battery, the original is supposed to be a KX-A36A. I measured the voltage on it, and had reversed polarity!
Turns out that the connector has two separate plugs and it's possible to plug in the battery with reversed polarity. There's no indication on the headset where +/- is supposed to go.
So the question is, was the battery wrongly plugged in or is the wrongly put in transistor causing the reversed charging polarity?
The battery is now slowly discharging on its own.