We wouldn't need any codebreaking techniques; we know their encryption method. I suspect feeding the papers through OCR would be the most time consuming part. I doubt decrypting would take any longer than SSL.
Am I missing something?
In short, the Navy, specifically U boats, used a non-standard, stronger crypto (in this, more rotors) Enigma machine, called M4. They formed their own Enigma network called "Triton" which the allies called "Shark".
Due to their hardening, and less message sent for traffic analysis, these machines were "safer" against cryptanalysis.
In any case, there are now (finally!) software implementations that are significantly faster than the reproduced hardware Collossus, so even if I'm not fully right brute-forcing messages should be viable.