Breaking a XOR cipher of known key length?
After more time on this challenge, here is what I've done to solve it.

Divide the text by N columns, where N is the length of your key.

Run frequency analysis on each column. Given enough data, you will most likely get a character that is present far more often than the others. Since my text was in french, it was the letter e.

For each column, find one instance of the value that is supposed to be an 'e' and XOR that data with the letter 'e' to find the Nth letter of the key.

Given enough data, you will find an 'e' for every column, giving you all the letters of the key.
Try decrypting the text with the key, and voilà, you get the original text.

Once I've made up this way of solving the puzzle, I found out that it was a really common way to solve a XOR encryption, and that tools existed that could do it for you far faster than manually.

For example, xortool : https://github.com/hellman/xortool

xortool -l 8 input.bin -c65


-l is the length of the key, -c is the hexadecimal value of the most frequent letter in the language you think the text is in.

That command is near instantaneous.