You arrive at a dining area with two long dining room tables. There are a bunch of potions and other stuff that you can find here, so scour what you want before leaving the room through an exit on the other side.  It appears that you have reached a dead end, but Frea thinks this is impossible and that there must be something more. She suggests you look around while she checks the dining area. The rooms adjacent to the dining area have shelves with some more potions, elixirs, and other alchemy related items.

Explore the corridors and carefully search along the walls, to find a platform with a handle on it:

Handle in Temple of Miraak Dining Area

Activate the handle. Frea reports that a secret passage in the dining area just opened up. Go back to the dining area  The secret passage is in between two bookshelves. Go through the passage, then through another door to arrive at a room with some statues and another handle. Frea says she does not recognize this statuary, and that you passed by a few of them earlier, but they are becoming more frequent as you get further in, and fears that they might come to life an any moment.

There are some ruined books in this area, you can take them to sell or for use at an Atronach Forge. Activate that handle to reveal a spiral stairway underneath the floor. Go down the stairs, there are some shelves one floor down with lesser soul gems on them if you need any. Go down a similar second set of stairs and arrive at another weird chamber with flames coming out of some sort of jaws statue. Frea remarks that it is eerily quiet and suggests you be on your guard. Activate another handle. The floor appears to move, and another passageway is revealed.

Go down yet some more stairs and arrive at a very large chamber. Boulders will fall from the ceiling above, so try to evade them. There are some Draugr, Skeletons, and other undead here. In particular, watch out for the Draugr Death Overlord which can be pretty tough to beat.

when you have cleared this area out, loot the chest for some potentially valuable items (sometimes even Stalhrim Items such as a Stalhrim Shield of Dwindling Frost). On the wall behind the chest (on the far edge opposite where you entered this chamber) there is a chain. Pull the chain to cause the section of the wall in front of you to spin, revealing a secret tunnel behind it. The tunnel will take you further downwards in the temple.

Follow the tunnel to arrive at a chamber with a pedestal containing a Black Book on top of it.