This is a relatively short mission that consists almost entirely of backtracking.
You start outside the Sunken Opera House, where you left off after Nevan. Run to the ledge nearby and grab the Stone Mask, then go back through the Sunken Opera House. In the Limestone Cavern, there are again a group of optional demons; kill them or skip them, and make your way back to the Subterranean Lake.
Now that you have Nevan, you can destroy the Combat Adjudicator you couldn't break earlier (Combo I - Slash - Combo II - Jam Session - Air Slash - Combo III - Feedback - Crazy Roll, start all over). Note that the melee attacks of Nevan are hard to raise the rank of the Combat Adjudicator, but with a Lv.2 or Lv.3 Swordmaster Style, the job can be done pretty easily. If you haven't already done so in Mission 9, you can pick up the Spiral gun, too.
There's a statue that you can examine between the two doors on the other side of the map; using the Stone Mask will raise a bridge that leads
to the next key item, the Neo-Generator, as well as the Secret Mission "Flight of the Devil". This mission requires several runes of Devil Trigger, as well as Nevan and the Air Raid ability, in order to obtain all Red Orbs in the Leviathan's Stomach map within the time limit.
Backtrack further to the Rounded Pathway, where a new type of demon, the Dullahan, introduces itself; they are essentially a possessed suit of armor and can only be damaged from behind by striking the red gem. This is easy to do with this passive variety of Dullahan, since these will literally only float in laps up and down the map without attacking (unless you jump while near them, they have an anti-aerial attack). Touching them from the sides or front will still damage you, however, since their shields are spinning sawblades. Eliminating these will open the gates and allow you to proceed again.
There are, again, Enigmas in the Underground Waterway. The camera trick described earlier works for the first half of the map; just run toward the edge of the ledge by the door to turn the camera away from the ramp, then run up (don't press Lock-On or that turns the camera back toward the Enigmas) without worrying about being shot.
There's nothing new in the Subterranean Garden (though you may wish to adjust your arsenal for the next map), so pass through to the Provisions Storeroom.
You will encounter the white Arachnes along with their bigger counterpart (green). Defeat the Arachnes and proceed back to The Rotating Bridge.
Approaching the door on the opposite side will trigger a short cutscene where Dante kicks the Neo-Generator into the slot above the bridge in order to power and turn it. Go to the big yellow door to go to where Leviathan crashed, double jump to the wall to find a Red Orb crystal. Jump above the Leviathan's eye to receive a hidden Red Orb cache.
Cross the new bridge to the Marble Throughway, where a familiar figure is lying on the ground. Approaching him will end the mission.