S P O I L E R S O B V I O U S L Y
The first thing to understand is that what the chapter is: a brief outline of this famous Duergar city, and a short adventure (the Whorlstone Tunnels). There are no real adventures in the city proper, if you don't count isolated random encounters.
The main problem is that the module never tells you it offers one (1) solid lead, for the party to find the actual adventure.
That is Drokki the Derro, down by the docks (at the Blade Bazaar, to be exact). You essentially follow him (or capture and scare the 
Before that, you need questgivers.
The only reasonably fully realized quest is Captain Errde Blackskull's, but as written this requires the tired tropes of "heroes being captured" and "outsiders getting preferential treatment" and "outsiders considered worthless, which is why we shower them with exciting quests and good loot". The good thing is that, unlike every other questgiver in town, Errde actually tells the PCs what they need to know: there's this derro called Drokki, that Errde believes is somehow connected to corruption within the Council of Savants, where they might find him (Blade Bazaar or Westcleft District), why the PCs might succeed where the Stone Guard hasn't, and what's in it for both Errde and the heroes (favor with the King; Errde's gratitude and/or safe passage)
The other quests are variations of this one, since the only real lead is Drokki. The problem is that the info is so scattered it's very hard as the DM to piece together the puzzle for the benefit of the players playing the module.
Errde actually has a second quest, called "search for corruption". But this is not a real quest, as in an adventure with a beginning, middle and end. There are no connected clues. This only makes sense after absorbing the entire chapter: then you realize this is a clumsy way of rewarding the players for rooting out the various bad guys they are about to root out anyway. This should have been worded more like "and by the way, there are rewards for rooting out evil, just so you know - be proactive" rather than an independent quest.
The "random" encounter with the two-headed Stone Giant. It's actually important, and should not be left out, though you won't know it until you've read the Whorlstone Tunnels section completely. It leads to an audience with Stonespeaker Hgraam. He provides nuggets of information for the far future (as in "many chapters of OotA later"), and actually doesn't provide a quest for the PCs. If you don't plan for the heroes to be given a reason to follow Drokki any other way, consider having Hgraam do it.
Then there's the Keepers of the Flame. They might "snatch" captured heroes, taking custody of the prisoners and instead lead them to Themberchaud's Lair. Where Themberchaud himself plays the same trick on the Keepers, taking the heroes for himself
The quest here is Drokki. You see, he's running errand for both bad-guy groups. So this is essentially the same adventure. You should probably not give out both this and Errde's quest, since that's lame.
Then there's the corrupted guard, Gorglak. Problems: when you read about him, the adventure fails completely to tell you he plays a part in the Grey Ghosts' operation. Also, chances are high the adventurers have never met him before when they complete the Whorlstone Tunnels, since he's stationed far away from the Darklake Docks. I suggest you simply make Gorglak the first patrol officer the PCs fall foul of, so the introduction is made.
Then there's Werz Saltbaron. I honestly don't know what his deal is. This chapter offers several pointers to Blingdenstone, and Werz provides one. But what the connection between Werz and Gorglak is, I can't say.
And there's "Ylsa's Deal". When I read this section I flayed about helplessly: where's the quest? What should the PCs do? And what is the solution? No answers are to be had - unless you catch a sentence at the very end of the Whorlstone Tunnels section, where a special coin is mentioned (with no reference back to Ylsa, naturally). Apparently the author once thought Ylsa could too be the heroes questgiver, providing information about reaching the surface in exchange for information on how "the derro gets access to surface coins". Problem is, the party needs to just follow any derro, and happen to choose Drokki. Also, as far as I can understand, the cultists get the surface coins "because magic", so this really never leads anywhere. And this would be fine, if the text actually said so anywhere.
Thank you for your Encounter Map. It is a real shame this chapter provides zero organisation. It must have been written by a mad Derro.
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