To defeat Udo Jin-e, go to the Bazaar and buy about 10 Storm Shields. Get the 90% variety. Stock those in your Sideboard.
Now stock your deck with as many Balefrost as you can, along with Ice Trap, Blizzard, Ice Wyvern, and Pixie. If you have Elemental Blade (which you can get from Niles for 1 training point), stock that, too. You might also put in a few Tower Shields.
The first spell you put down should be Balefrost, which will give all future Ice spells a 35% boost. Then you want to cast two Blizzards, which will get rid of the minions. You don't need to put Ice Trap on the minions. After that, you'll focus on the Udo Jin-e.
When you don't have the pips to cast Blizzard or Ice Wyvern, you should be drawing Storm Shields from your deck and using them. Also, once you have Balefrost down, discard all that you see after that, as you can only use it once. If you need to heal, use Pixie. If you have plenty of shields on you and don't need to heal, put Ice Trap on the Udo Jin-e.