The algorithm isn’t broken, the definition of a good encryption scheme is that it can’t be decrypted unless you have the key, in this case the algorithm is to substitute each letter with a lookup into a document. The non-public key is the name of the document not the document itself.

Ostensibly the author decrypted the the second document by doing a brute force search of the key space (that is the set of documents available at the time). This is functionally the same as aes - just a much smaller key space.