An update. It was the battery after all that got killed in the fall.

TL;DR

My phone suddenly shut down in the middle of playing a game. Presumably the battery got finally drained. But then it would not charge. I opened it up. Took the battery out and put it back in again. Tried three different cables. One, from a USB hub, would do nothing at all. Other two, one straight from the laptop's USB connector and one from the wall charger, also looked like nothing was going to happen, but after a while (about a minute, I reckon) the phone booted to the charging animation, showed the battery empty, switched off, then continued looping like that.

That's it, I thought. The phone is dead, like Trzyzet said. But then I had a brilliant idea. The sign of a true genius. I replaced the battery only recently. The old one puffed up so much that it opened the back cover. It seems like ages, but it was less than 2 months since I got the new one. I may still have the old one lying around! Rummage, rummage... YESSSSSS! Found in the discard box, waiting for recycling. Popped it in and... voilá! It works! Status 57%, increased to 60% by now.

Phew, that was a relief. A new battery is a bit less hassle than a whole new phone

So you're in the lucky 10%

Not really. It was a hardware fault after all. Just not on the main board but in the battery.

I suspect the battery connector. The battery was violently yanked out so it is the most likely part to have suffered. But the fall was too short do damage anything else (well, except the screen, but the screen survived). But I am too lazy to investigate further. A new battery has been ordered

Hopefully the last word on the topic. The new battery arrived, and here is a side-by-side photo. The one on the left definitely looks like the connector is borked.

I don't know how many last words one can have but here is another one

Now that I have three batteries (1-the old puffed up one, 2-less than two months old with a broken connector and 3-brand new), I decided to swap the control board from #1 to #2. After stripping the decorative (really, it has no other function) wrapping, it seems like a trivial operation: