Elitebook 8460p Win 10 x64 - battery issue + BIOS update

[123viva456]

Hello, 

I have quite a long and complex issue, sorry in advance.

I own a HP Elitebook 8460p (x64)  bought second-hand in March 2020 (Windows 10 pre-installed).

The battery has always been quite weak.

But a few weeks ago Windows has stopped detecting a battery at all, and the laptop turns on and works only when it is connected to A/C power.

Yet the battery is still in the laptop, Windows just doesn't detect it ("Battery not detected"):

I had supposed that the battery had died and bought another one: a GREENCELL CC06 Notebook Battery 6360b

Which was clearly presented as being compatible with this laptop.

New problem: when I put in this new battery, the laptop does not turn on at all, even when connected to A/C power.

But when I put back the old battery (which Windows still does not detect), same as above, the laptop turns on and works only when I connect it to A/C power.

So I supposed that the issue originates either from the new battery, or from my computer itself.

I tried to update the BIOS, as it seems old ("Hewlett-Packard 68SCF Ver. F.60, 12/03/2015" ; SMBIOS Version - "2.6") and I read online it could solve my problem.

New issue: the BIOS update downloadables are not available on HP website for this OS and this laptop: 🙃

Nevermind, I have downloaded the latest version of BIOS update for Windows 8.1:

But when I went through the BIOS update process, this is the error screen I got: 

Translation: "The computer is not plugged into the external power supply. Plug in the external power supply and select Enter and continue."

So BIOS update seems impossible due to my battery issue, whereas I was trying to solve my batterie issue with BIOS update.

Can anyone help me on this?

I precise that I have tried the purge process (turn off, remove external devices, remove battery, hold Power button for 40sec, re-insert battery, turn on).

It hasn't worked. 

Thank you in advance!