kata discription is shit

you have VALID_WORDS set
try console.log(VALID_WORDS)

That condition is necessary but not enough. You can be sure that with less than 17 you can't have unique solutions, but as you see, with 29 givens you still can have multiple solutions.

The first solution is not unique, this one works too:

[[8, 2, 1, 5, 7, 4, 6, 9, 3], 
 [3, 4, 5, 2, 6, 9, 1, 7, 8], 
 [7, 9, 6, 8, 3, 1, 5, 2, 4], 
 [2, 5, 9, 4, 8, 7, 3, 6, 1], // In these 2 files we swapped 9 and 4 of the 3rd and 4th columns
 [1, 3, 4, 9, 5, 6, 2, 8, 7], 
 [6, 8, 7, 1, 2, 3, 9, 4, 5], 
 [5, 7, 8, 3, 9, 2, 4, 1, 6], 
 [9, 1, 3, 6, 4, 8, 7, 5, 2], 
 [4, 6, 2, 7, 1, 5, 8, 3, 9]]

Or it should raise an error in cases of: invalid grid (not 9x9, cell with values not in the range 1~9); multiple solutions for the same puzzle or the puzzle is unsolvable

I suspect the other may be the same case.

[[3, 9, 6, 4, 7, 5, 1, 2, 8], 
 [8, 7, 1, 9, 6, 2, 5, 3, 6], 
 [4, 2, 5, 1, 3, 8, 4, 9, 7], 
 [1, 6, 3, 7, 8, 4, 9, 5, 2], 
 [7, 8, 2, 3, 5, 9, 6, 4, 1], 
 [5, 4, 9, 6, 2, 1, 7, 8, 3], 
 [6, 1, 4, 2, 9, 3, 8, 7, 5], 
 [2, 5, 7, 8, 4, 6, 3, 1, 9], 
 [9, 3, 8, 5, 1, 7, 2, 6, 4]]

the kyu values are all over the place, I've seen pretty hard 7 and 6 kyu and really easy 4 kyu.

Read the note or the last post.
https://www.timeanddate.com/calendar/?year=1001&country=22

My guess is that you modified the input string instead of returning a new one ...

1000+ people solved it without a problem...

what I get is this

supose you receive 'AAAB', then all posibilities are (always based on the letters)

AAAB = 1
AABA = 2
ABAA = 3
BAAA = 4

if you get 'ABAB', all posibilities are (if I don't miss anything)

AABB = 1
ABAB = 2
ABBA = 3
BAAB = 4
BABA = 5
BBAA = 6

and the task is to return the correct index