This table shows you the exact time of the winter solstice and its date for various years. Note that the dates cycle between 21 and 22 December and the times are in 24-hour notation.
But it seems obvious that if we have a shortest day in a gradient progression the longest night will be the one just before or the one just after it -- more likely the one before it, since the solstice marks the point where days start growing longer in the cycle. There are only 24 hours in a full day cycle (rounded to the nearest whole hour). ANy time in that cycle not occupied by "day" must be occupied by "night" unless you want to complicate things with nautical twilight, crepuscule, and such middling zones.