I think you can find plenty of shops doing electronics repair in Paris (especially in Paris - somewhere out in the woods it would be a lot more complicated).
E.g.:
https://www.yelp.fr/search?cflt=electronicsrepair&find_loc=ParisI don't have any first hand experience with any of them though, so do your own due diligence before handing your device to any of them.
I could have a look at it too, but I am quite a bit farther up in the Ch'ti land, so you would have to drive the mess over to me. From the symptoms you are describing it is quite clear that something has released the magic smoke there, quite possibly some of the digital chips since the digital inputs have failed first.
Now, realistically - your speakers were sold for some 330 euro as far as I can see (it is an obsolete model). If there is some serious failure requiring major work or replacing a difficult to get component, repairing it would likely cost you more than the price of a new set, certainly more than the current value of your device. So don't be surprised at the price quote you will get for the repairmen. You may be better off buying new speakers.
Re your experience with Logitech - that's to be expected, unfortunately. This is consumer hardware, not designed nor intended to be serviced. It is designed to last for the mandatory warranty time and then when it inevitably breaks to be thrown away and replaced. So the companies don't have any repair service anymore - it is just not worth the expense.