Chosen Solution

While opening my Iphone 5s to change the battery, when doing this :

the LCD connector flex cable just disconnected from the LCD screen, as if it was not really sealed (maybe due to an old drop of my phone..) I don’t even know where to “plug” it. Is there a way to fix this easily ?

Sorry, but it sounds like your LCD assembly is toast. You will have to get a new LCD assembly. Link: iPhone 5s Screen If you want to be a little cheaper: iPhone 5s LCD and Digitizer