The really important point with soldering is the joint must be physically
strong before you solder. DON'T rely on solder to provide both strength
and conductivity, it wont !!.

I've just finished soldering the joints and the power is now back on (hence
my computer being up and running...)

Here's what I did. Please don't say this was wrong (this is 2.5mm core by
the way):

1) Stripped around 1in of wire on each cable
2) Twisted them together using pliers (the best I could - often it was the
case that one wire was tightly coiled around the other)
3) Soldered them so that the solder "flowed" into the joints between the
cables (visually, this was not unlike what happens when welding pipes using
presoldered joints)

To be honest, the concept of soldering cables like this strikes me as odd.
Despite my best intentions (and those of anybody attempting this job), there
are places where the solder will be providing the conductivity. I simply
can't guarantee that copper touches copper all the way through.