No, the second cast of Distortion doesn't trigger Lich Bane to empower your next auto attack.
Watch the Spellblade buff on Leblanc's buff bar - you may need to watch it a few times. When you see the Spellblade buff your next auto attack is empowered, and the buff lasts for a number of seconds before you lose the empowered attack (you can see a countdown effect).
Every additional ability used without making an empowered attack refreshes the Spellblade buff. This means that if the Spellblade buff is not refreshed by using an ability, then that ability does not activate Spellblade.
If you watch the video, you can see Leblanc jump forward with Distortion, granting her the Spellblade buff from Sheen. When she uses Sigil of Silence shortly after, the buff is refreshed. However, when she returns by re-activating Distortion, the buff is not refreshed.
Since the buff isn't refreshed on the second cast of Distortion, the second cast of Distortion will not activate Lich Bane. I would safely guess that this also applies to the second cast of Mimic: Distortion.