Nchuthnkarst is the largest public dungeon in the Greymoor expansion (try saying its name fast, I dare you). You can find it in Blackreach, at its center. It is so big, it has its own map you can inspect on the global map. The closest Wayshrine is Dusktown, located in the southern part of this cave system. There are no great lifts from Western Skyrim that can take you here. On the other hand, if you use any of the three greater lifts above, you should get the Dusktown Wayshrine unlocked automatically. Otherwise, you’ll have to make it here by foot from the other sides. The dungeon is just south from Greymoor Cavern Great Lift, east from the Miner’s Lament ritual site, southwest from the Gloomforest ritual site, and north from the Dwarf’s Bane ritual site.
There are a lot of lamp posts near the entrance of this dungeon, and even without looking at the map, you can spot that there is something going on in here. The Tones of the Deep is the name of the quest given by Kireth Vanos, who is next to the dungeon entrance. Press the lever to go even deeper and reach the Nchuthnkarst area.
From the entrance, you’ll need to make your way to the first door to Nchuthnkarst Control Room. Considering you are alone in here, you’ll have to deal with a group of enemies. They will definitely follow you, blocking you from going further until they are dealt with first. In the next room, you should go forward to the next door and into Nchuthnkarst Complex, but if you go to the right here, you can see your goal in the distance – a broken bridge on the other side. The skyshard is at the end of said broken bridge.
To reach this broken bridge, you can go around the central room, following the path over the bridge to the north, going under the bridge from its western side, or good old jumping towards it, hoping you’ll survive. There is a large mushroom east of this bridge which you should aim for if you decide to jump down. From here, whichever of the paths you take, simply follow the road and aim for the broken bridge section, where the shard is hidden on the edge, overlooking the crescent-shaped pool.