The thermostat is contained in its own housing so you will have to change the whole thing. Why don't you take the unit off & test it, cold water will not run through, very hot water will cause the thermostat to open & permit water to pass through, if cold water runs through then you will know that the thermostat is stuck in the open position. Do cars wintering in Iceland have radiator jackets fitted to stop icy air from reaching all of the radiator? Try asking dealer if one is required, & can be supplied.
Is it true that the Icelandic Govt. are trying to get smoking banned throughout the whole of the country? Good Plan !