This paper introduces a machine learning (ML) approach for cryptanalysis of the ciphermachine Hagelin M-2091. For recovering the part of the secret key, represented by the wheel pins, we use Artificial Neural Networks (ANN) which take as input the pseudo-random displacement values generated by the internal mechanism of the machine. The displacement values can be easily obtained when ciphertext and plaintext are known. In particular, we are using several distinct ANNs, each recovering exactly one pin. Thus, to recover all the 131 pins, we utilize 131 model seach solving a binary classification problem. By experimenting with various ANN architectures and ciphertext lengths, ranging from 52 to 200 characters, we identified an ANN architecture that outperforms others in accuracy. This model, inspired by the architecture by Gohr used for attacking modern ciphers, achieved the following accuracies in recovering the pins of the first wheel of the machine: approximately 71% for 52-characters sequences, 88% for 104-characters, 96% for 200-characters. The first wheel has the largest size and hence represents the most complicated case. For the other wheels, these accuracies are slightly higher. To the best of our knowledge, this is the first time when ANNs are used in a key-recovery attack against such machines.