Your first mission using The Legendary Devil Hunter Dante! Since that happened, Dante will inherit The Dead Weight Nero's Vitality Gauge, Orbs and fully refunded Proud Souls. Use these to power up Dante. Though, you probably want to get Swordmaster to Level 3. Besides that, buy anything you want.
Ascension Chamber & Meeting Room
You start The level in the room you fought Dante in Mission 10. If you're new, you might want to take advantage of the Scarecrows that spawn in this room to check out Dante's abilities. Otherwise, you can take the opportunity to fill the Disaster Gauge if you have Pandora, or just ignore them and exit the door that leads to the meeting room.
When you exit, you will notice Secret Mission 7. Attempt it if you want, but then continue farther into the meeting room where you will be introduced to a new enemy - Mega Scarecrow. They may have more health than the normal Scarecrows, but in general they are just as weak as their arm and leg counterparts. After you defeat the Mega Scarecrows, head out the door to the elevator room, and remove the Wing Talisman from the elevator. Moving forward from here starts a 10 minute countdown. This should be enough time and then some, so long as you don't dilly-dally, so don't panic.
If you have a full Disaster Gauge on Pandora, you can use Argument to float across the chasm. (If you have trickster level 4 you can DT, stinger, then "triple jump" and double sky star.) If this is your first time through the game, however, you have no choice but to jump. Last one down is a rotten egg! Pick up the gold orb sitting in the far corner, then book it to the other side of the room and hit the mechanism to call the elevator. Upon entering the cage, you will fight some Frosts. Defeat them to continue. The cage will rise and let you out at the Experiment Disposal room. There are some Bianco Angelos here but you can just skip them. You probably should whack the Red Orb cache before you go, though.
The next room contains the obligatory Pointless Spinning Blades of Death—this is an evil mastermind's lair, after all, (or maybe they're just some sort of elaborate fan). Some Mephistos and a Faust will appear and block your path. The Coyote-A seems to be more effective than Ebony & Ivory on these guys, so use whatever technique you can to dispatch them quickly. After the battle, activate the Chrono Slicer, and just climb the blade to the security corridor. There, you can use the Chrono Slicer to dodge the laser beams, or just hover over them if you have sufficienct Disaster Gauge on Pandora. There is a secret mission here; if you haven't done it already, well, too bad: Dante can't do this one. After you get through the security corridor, you will face some Bianco Angelos in the Grand Hall. Kill them to dispel the barrier. From here, you can just waltz right out if you like; but, if you have time after the Angelo guys—about 3 to 5 minutes—you should probably take a detour to the Gauntlet room to pick up a Blue Orb Fragment.
Go through the stairway/hallway on the left of the room into the room where Nero first obtains the Key of Chronos. There will be Gladii here, but fortunately you can save yourself time by activating the Chrono Slicer and running past them. Once you're through the lasers, you'll find yourself in the Gauntlet. Grab the stash of free red orbs in the center of the elevator shaft-like hole and continue to the far side of the room to find a Combat Adjudicator for Dante. You only need to get your rank up to B, which shouldn't be hard at all with Swordmaster level 3 or some clever switching of Styles. After you're done there, turn around and leave.
You should have enough time to run back to the Grand Hall, again ignoring the demons that spawn in the Key room, and out the front door. A cutscene will trigger just as you leave, during which the Headquarters will be destroyed while Dante watches smugly. Cross the Gran Album bridge to finish the mission. Don't forget to destroy all the little things in the stable-like space on the bridge!