Did you ever deworm her?
I just remembered someone with a little Salmon Faverolle chick having similar problems and deworming did the trick: on the 5th deworming day it showed that a nasty tapeworm was causing her problems and after getting rid of it she thrived.
Living in France you should be...
Sometime a deficiency of the parent bird shows in the chick.
Offer some boiled egg mashed into little pieces, sick chicks will also often take some millet and fine rolled oats as it is small and easy to swallow.
Maybe check her also for a long hair or other kind of thin string/fibre that might...
Could she have gotten bit or stung by some poisonous insect or critter?
Regarding the vitamin supplement, it can take days to weeks to see improvement depending on which vitamins were deficient.
If you see her already getting better after one day only, you might stop the daily supplementation...