In reply to mountain.martin:
> I had something similar with my Dell laptop. I bought a replacement but when I came to swap it out ... found a connection cable had somehow come loose and that was the only problem.
That would be my first suspicion if the battery had been fine and then suddenly went 100% u/s. To the OP: rather than buy a new battery as a first resort, get the back off and check the connections on the existing one.
As for the battery being non-removable, IME laptops are reasonably straightforward to get in to provided that you can find the screws (at least one is usually hidden under a tamper-proof label, other can be concealed by body colour plugs) and you have the right screwdriver bits (eBay is a good source of odd-shaped bits - searching for "security bits" can often turn up the right ones). Once inside, the service items like batteries, hard drives, screens etc usually have plug connectors which are relatively easy to disconnect and reconnect. They're not like smartphones that need to have the cases prised carefully apart without breaking any of the clips, often after being heated with a hairdryer/heat gun to soften the glue, then re-sealed when you've finished the internal microsurgery - which all too often involves delicate soldering that is easy to mess up.
Push comes to shove, a reputable high street laptop repairer should be able to fit a new battery for not that much more than it would cost you to do - especially when you factor in your time that you could otherwise have spent climbing....