The fix was simple, but took me a short while to find on the net, hence this post. Written in Layman's terms, should someone non-techie find this post.

Unplug the laptop.
Turn it over and remove all of the small screws. Keep them safe.
Gently 'pop' the edges of the laptop's cover open. They have tiny clips around the outside.
As you remove the cover, do it slowly. There is a cable between the power supply (not the battery) and the motherboard that is quite short, you could damage it if you pull the cover off with gusto. You will need to leave this plugged in to follow a step below, so leave it connected.
Locate the small connecting wire between the battery and the motherboard. It's located between 6 and 7 on this image
Unplug this connection. It's small and delicate, so be gentle. Treat her like a lady.
Contrary to what you may have been told before, plug the power supply back in to the laptop and turn the power on. Leave the laptop itself turned off, just supply it some power. Don't touch anything inside the laptop and don't electrocte yourself, please. This step is to get the laptop to pick up the fact that the battery has been disconnected.
Wait around 30 seconds. The LED for the battery indicator should change to red. The Chromebook is stating that it can't find the battery.
Turn the power back off again.
Plug the battery connection back in to the motherboard, the one that was pictured above.
Put the cover back on, gently clipping it back in to place.
Plug the laptop back in once more and turn it on. Boot it up.
Hopefully, it should now be charging, as it's picked the battery back up again. If so, put your little screws back in and off you go.