Head up the stairs and turn to the right. You will come to a hallway way. Toward the end of it, you will see a few Vandals and a number of Shanks. Quickly clear them out. There are some more on the way. This time mostly Dreg. Check over everything and head up the stairs and turn to the right.
At the top of the stairs you will quickly encounter another melee Vandal. Move back if you have to and take them out before they can get through you. Continue to the right and you have more Dreg. Head down the hallway to the next room. Be ready as you this is a large area and there are no shortage of foes.
The landing just below you has a few Dreg on it. They are not the concern. Kill them and more will come in from across the room and a Vandal. Look to the left after you deal with them and you will see a Servitor, another Kell's Guard (Ranged) and a few more Vandals. You will want to get through all of them. On the far side of the room you will see a door. This is the entrance to the Throne room inside the ship.