Irelia is great as assassinating champions with no defenses. through sheen and You had no defenses.
Also, looking at the game, you didn't have Trinity Force. You had Phage and Sheen but you didn't get the additional power from completing it.
Also, while Irelia may have had a worse KDA, she was still about equal in gold with you through farming. Considering Irelia's damage is typically more bursty than Udyr, you lost when you didn't have enough defenses to survive long enough to apply your damage.
Her Mercury Treads were pretty important since they reduced your stun duration and gave her MR to handle Phoenix Stance and Devourer's damage.