Is my laptop battery broken or what could it be?

Since lately I can only boot up my laptop when connected to the power supply. No matter how long I let him charge, below is always "0% remain (mains operation, is not charged)".

Would it be of any use if I replace the battery or is the damage somewhere else?

Yes. Either you buy a new mostly 20-40 euro or use it only via power cables what would go, if you want to use it like a laptop again you have to buy a new battery maybe you still have warranty sometimes they make that out of goodwill.