Across 1,336 ranked games on patch 16.21 in Master and above, Kayn mains win 52.8% of their games in the jungle.
Their most common build appears in 20.9% of those games and wins 56.3% of its 279 games.
It starts 
Gustwalker Hatchling, completes 
Voltaic Cyclosword, 
Axiom Arc, 
Edge of Night, 
Serylda's Grudge and 
Guardian Angel in that order, and takes 
Ionian Boots of Lucidity.
Three other builds are played often enough in the jungle to be measured on their own.
In the jungle, Kayn mains fare best against 
Ekko (won 77.3% of 22 games), 
Nocturne (won 80.0% of 10 games) and 
Talon (won 62.8% of 43 games).
Their hardest matchups, the champions that counter Kayn, are 
Hecarim (won 41.0% of 39 games), 
Nunu & Willump (won 33.3% of 15 games) and 
Master Yi (won 40.7% of 27 games).