Heroic LK requirements

For a bit of info on our position. Our guild just formed 2 months ago on alliance Faerlina. (We are all aware of the server pop and don't want to switch) We were apart of another guild for a while, but the leadership was poor so we switched.

To the point. Last week we 11/12 cleared heroic ICC. It took us about 2 hours and we got a few pulls on Heroic LK. We want to save time on clearing by only doing the wing bosses on heroic this week, but we are unsure on if this would unlock LK.

So what is required to unlock Heroic Lk? Do we have to full clear heroic again? Can we just full clear on normal since we already did it on heroic once?

Any info on getting to Heroic LK faster would be appreciated.