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No, but not because of the shooting/hitting part. It's because the spell is a targeted spell, targeting the foe, and such spells require being able to see the target. If you used the spell on them, THEN they went invisible, it would be perfectly valid to continue using its effects as you hit them.
EDIT: For fun, get a spell-storing weapon to bypass the targeting restriction. Hit an invisible foe? Free action to use the stored Instant Enemy on them despite the invisibility.
No
Instant enemy targets.
You must be able to see or touch the desired target to be able to legally select it as a target.