Denmark does not issue any non-passport document that is recognized as an "identity card" for EU purposes. The rules about traveling in the EU on an identify card are aimed at citizens of member states that have mandatory government-issued photo IDs for domestic use; these IDs are then also good for intra-EU travel under the freedom-of-movement directive.
Denmark does recognize "does not have time to get an ordinary passport" as a reason for issuing a temporary passport ("provisorisk pas") covering only the period of intended travel, if the person's previous passport expired more than 3 months ago -- but the travel plans here are cutting even that very tightly, with a planned departure on the first working day after Christmas.
These temporary passports are recognized by all EU countries and certain other countries. (The Ministry of Foreign Affairs attempts to keep track of which countries recognize them at http://um.dk/da/rejse-og-ophold/rejse-til-udlandet/pas-og-visum/).
Even though Danish passports are now usually issued by municipal authorities, the police stations at Copenhagen and Billund airports can also issue temporary passports for super-urgent cases. Your sister should get herself to one of those places ASAP, bringing as much documentation as she can scrape together, including at least her old passport and her birth/baptism certificate, as well as a passport photo.
(For all I know, they may end up telling her to apply for an emergency passport at her municipality first thing in the morning Tuesday instead, but it's better to pursue both avenues).
