Here are some hints for that repair job coming from a perspective of someone with less experience. I have never used a Multimeter before (I got one specifically for this and some cheap cables and tape to connect a battery) and worked with batteries only so far as to put them into things.
Some things that nobody has mentioned about fixing this yourself is that the plastic covering the metal is really tough and annoying to deal with. (and it's best to get rid off all of it)
Someone said to use a razor blade, which I did. I probably should have switched sooner to maybe a paper knife, which in hindsight might have been better, as the plastic and even the outer leather skin took me ages (it was a leisurely project, where I didn't really insist on using my brain much and being terribly efficient is my only excuse).
I also used a 9V battery for adding (?) the voltage, and found that the first tenth of a volt go really quick and it really hits diminishing returns fast.
I started at about 1.6 V on each battery.
In order to get every battery up to about 2.7 V (2.3 V wasn't sufficient), I actually had to let it sit for an hour or for each battery (9V is probably not the best Voltage to use, but I only had 1.5V lying around, which is probably too low, if I had to do it again I'd try 5V).
Also take care of not mixing up the connectors or you'll actually substract some Voltage (which I did, but that can be reversed).
Another thing is that when you open the back and the connector is facing away from you and you look at the four batteries in front of you, that the first battery will always have the same Voltage as the third one, and the second one will always have the same voltage as the fourth one (maybe that is something super obvious for someone who knows things about electricity). So you only need two charge two of the outer ones and don't have to bother with the others.
When putting the thing back together adding the metal plate back, screwing all the two dozens little screws back in is a waste of time, but 3-5 of them are bit shorter and wider than the others (it's subtle), it's probably best to mark their position and have them in.
So I hope this encourages people to try this, if they find themselves out of warranty, like I did.
I imported mine from the US to Germany and got a full refund and got to keep the thing, so I'm
fairly happy, that I did the repair. Just having a needlessly thick, non-glowy type cover would have
been lame.