It stops the process with all four LEDs lit but what is supposed to happen is that they all go out when fully charged.

Hi Daisy, it is normal the led lights stop when the battery is fully charged, and when the battery power is higher than 95% before you charge it and you want to charge it, you need to power on the battery by this way: press once and press and hold for a second then place it in the charger to get fully charged.
Is there any problem with your charging the battery?