Recovering key length from transposition block cipher ciphertext
It really depends on the key schedule.  IE, how is the key operated on to find the permutation to be used?  

Absent that knowledge, really the only way to know is to launch searches in parallel at different key lengths, and the one that finds the answer first had the right key length.  Unfortunately anything more clever than that will utterly depend on how the key is used to generate a permutation.  

Just a note:  As described, this cipher is terribly insecure; the attack is called parallel anagramming, and is not really more difficult than a crossword puzzle.  It breaks the instant the opponent has more than one block, assuming the same permutation is used for multiple blocks.  

Permutation ciphers also fall to even a single block of known plaintext.