I would say that your battery has went into deep discharge and that some cells are defunct. Your 13V is your alternator running and trying to charge up the battery, but if it is gone inside, which looks like it, then you'll never chargevit up.
The battery is probably 5 yrs old and depending on its lifestyle it is not uncommon to change or a new one. The one I will say is that the Varta E44 is a higher spec unit than the one that's in the car at the moment and will be a huge difference in power delivery and running.
It has been well documented here on ASN, do not hesitate and get one!
S