I figured it out by means of buying a spare battery, the only left spare batteries for this model in the world were in New Zealand. Lucky I had a friend who lives there and bought it for me and delivered it to me in Europe.
Charging it as we speak I will test it later on. But reading the manual thoroughly again about the charger docking station, it says "the charging indicator will come on..." But I don't see any indicator lights on the charger station (5KCL12CSOB).
Did you see any indicator lights on it?
It makes me feel that actually my battery wasn't broken, but the charging station is.... I'm charging the new battery now, so if that fails to recharge, then it must be the charger station.
If that's the case, do you know if there is a different charging station available, maybe even by KitchenAids own brand, that can charge these batteries? A different model I mean, one that has indicator lights? Or even an after market one?