HP ProBook Elite laptop is randomly failing to turn on?

My daughter's HP ProBook Elite laptop is randomly failing to turn on.
The battery might be faulty, because if I unclick it from the laptop and take it off, the laptop will sometimes start up. The battery charges normally though.
We've not found a reliable way to turn it on when it does this. We disconnect the power and reconnect and then press the power button and eventually it starts. Not found a repeatable way of doing this which might point to the fault.