Dane is just an anagram of Dean. The "fact" she was a "Dane" is just an inside joke at John Dean and his cooperation in the bringing down of President Nixon. Jokesters, think of them as presidential merry pranksters, put out the word she was a "Dane" to alert any "whistleblowers" that no matter what they might say, there will be no Dean this time. If they try it, they will be shot.
Cheny and his henchmen think of John Dean as a loud-mouthed chick. And that is the source of the "she is a Dane" witticism.
It is all so easy when you just take the time to connect the dots.