Kirkenes sits inside the auroral oval — the band around the Arctic where the northern lights appear most frequently — and the dark winters from September through April bring regular displays. The aurora is not guaranteed on any given night (you need clear skies and solar activity), but a week-long stay in midwinter gives good odds of at least one strong show.
The best strategy is to get away from the town's modest light pollution: drive a few kilometres out on the Pasvik road, or take one of the guided aurora tours that run on snowmobile or by minibus. Tour operators also offer alert services — they call you at your hotel if the aurora kicks off after midnight. The Snow Hotel, being outside town on a lake, is particularly well placed.