I believe they rather want information which they cannot legally force her to divulge (otherwise see above), so they're trying their luck at intimidation.
Jail time does seem appropriate for non-compliance with a legally binding judicial order (I'm not endorsing or condemning the existence of these laws, btw)
Through the eyes of a person compelled to give testimony, we may reach a different conclusion. Consider a journalist who is granted immunity and asked to give up their source. They refuse and spend six months in jail, with few to no options to appeal for their freedom. This isn't theoretical. It happens occasionally.
The definition of due process can vary from the common definition of fair treatment under the law, to the view more often taken by lawyers and governments that it just means following the process of the law as accepted by the courts and respecting legal rights.
The reporter probably would feel as though they weren't treated fairly under the law, but the prosecutor would argue that they followed the law in full. I probably should not have used the phrase "without due process" in this case, since that will make it difficult to communicate with anyone holding views of the government or lawyers.
To make things more difficult, the very contents of the letter itself are considered a national secret. Typical methods for serving papers would needlessly jeopardize the confidentiality of such papers. So one would presume that these methods simply aren't in the cards.
Government can only compel you to hand over information for third party which you possess. They can't compel you to DO something, as demonstrated by FBI v Apple recently.
In case anyone who's not familiar with the Apple case is reading this: The government didn't actually wait for the court to rule that they can't (which might or might not be the case, though my understand is that they would've probably ruled in favour of Apple), but rather withdrew the case after a third party unlocked the phone for them.