And the letter, which was sent as priority mail, had been opened when they went to retrieve it...
IANAL, and can't profess to any knowledge whatsoever of Danish law, but opening a package clearly addressed to someone else without permission may be reasonable grounds for litigation.
Though to the question "what good will that do", you're right, it's not like new health records can be issued.
Depending on the details of what was shared and what ties them to an individual though, I suppose it might be possible to issue new IDs.
If anything, this requires a severe audit of the security practices of the affected organisations. Moreover, I think citizens of Denmark are entitled to know what information about their personal health records is leaked.