Mixing Autokey with Regular Vignere It seems to me that cryptanalysis of the autokey cipher relies on patterns being found in key attempts, easy when it's English you're looking for, but how can you do this if you're just getting further cipher texts?

How would someone decrypt a plaintext enciphered with an ordinary vignere followed by an autokey cipher, where the two keys are 10-20 characters long and random? 

Presumably any solution would rely on a computer, but I'd also be interested if there was a method that could be done by hand.

Thanks
It can still be done, because some letters of the English alphabet are used more often. Analyzing a ciphertext based on the frequency of appearance of some characters is fittingly called Frequency analysis.

This method of cryptanalysis gets better, the more ciphertext you have. If the plaintext is in English, then you would know, that the most frequently used letter would be "e".

Because the Vigenère cipher is quite an old method of encoding something it can be (and has been) broken relatively easy by hand. The security relies on the keylength and the "randomness" of the key. 10-20 random letters is actually pretty good, but with a very long ciphertext you can ultimately still break the Vigenère method.