Intuitively, I'd think dropping the xperia might have unconnected some hardware inside, most likely hardware linked to the wifi.
I watched the following videos to get more knowledge about what is inside that sealed water resistant phone.
From those videos, I planned I'd open my phone with an airdryer and small screw drivers. It's more sketch than what is suggested in the videos, but the phone is 2 years old already and I don't plan on keeping it many years longer.
So, I opened the phone and it was a scary experience. Hopefully, I managed not to screw up much of the glue setup. Like in the videos, I disassembled the battery and the bottom part of the phone in order to get to unplug and replug several components to make sure they are properly plugged.
I put the cover back on the phone and it sticked ok... From now on, I won't put it into water anymore, I can live with that.
Booted it back up and set it up. It works just as new.