You've got the right idea, but a little too glass cannon due to mastery tree and items. She simply becomes TOO squishy. I suggest Rylai's and/or Liandry's and some of the defensive tree.

And yes Lich Bane is perfect for a caster triggering it nonstop.

Iceborn is nice but Lich Bane is better for boosting your damage up. Plus you'll have excess CDR from Iceborn Gauntlet (5% from masteries + 10% from Eye of the Watchers + 20% Athenes + 20% Iceborn gives 55% CDR which is over the limit)

I prioritize three things in this build, maximizing your abilities' effectiveness through AP, CDR and mana regen. All in all I just find Lich Bane so much better at that than Iceborn Gauntlet.

uhm please don't go lich bane.
the spellblade part is super cool but its a midlane item, not a support item.
you need more utility not more damage. So, instead, go iceborn gauntlet. you will be surprised how much dmg youre dealing while slowing them too