Picks or axes, I imagine - stuff that can be used to break up ice.
Spears would be pretty common, too.
Elvanna's Winter Guard are armed with swords.
Lamashtu and Zon-Kuthon are the kingdom's main religions (one of the duchesses pulls double duty as Irrisen's high priestess of Zon-Kuthon), which may also influence weapon choices.
Lamashtu reflecting the kingdom's high population of monsters who are actually considered part of the kingdom, and honestly form the basic citizenry (since the non-Jadwiga humans are almost all serfs).