This is the quest that reads as broken and is not. It asks for two things and names only one, so a player who clears zombies around the workshop and watches nothing happen has usually done the work and missed the last step.
The short answer is that one battle counts rather than any battle, and that winning it does not tick the quest by itself. 
Jack has to be told.
The battle is the Market Square Purge, and there is only one
The fighting half is its own entry in the journal, called Plaza Battle, and its objective is a single line: win the battle Market Square Purge. No other fight on the map advances it.
On the map it sits under the Square label, north of the crossroads where the guards keep their camp outside the gate. Walk up past the nun and it is the icon on the far side of the wall.
It wants 22 defense power to start and fields two squads, which is double the single squad the sewer brawls ask for. If the Start Fight button will not light, that number is what you are short of, and barricades and towers is where it comes from.
| Squads | Two |
| Defense power needed | 22 |
| Reward | 6 Zombie goo and 4 Town Gratitude |
| What the win opens | The door into Jack’s workshop |
| On the map | Under the Square label, north of the crossroads gate |
The Eastern Districts purges, the sewer brawls and the floodgate battles are all separate quests with their own rewards. None of them touches this one.
Two things prove you fought the right one. The Market Square icon greys out on the fight map, and the door into the workshop becomes usable. If neither has happened, the battle is still waiting.
Winning it does not tick the quest
There is no completion check on the quest itself. It closes when you talk to Jack, and the journal line has been telling you where he is the whole time.
I need to clear the zombies around Jack’s workshop, and then bring him the good news. He’s staying at the Town Warehouse.
Journal, Jack’s Back in Business
So walk back to the Town Warehouse rather than to the workshop. He is not standing in the building you just cleared, which is the other half of why this one feels stuck.
The conversation that follows is worth having for its own sake, because he is not remotely grateful about it.
You should have kicked those undead out of my workshop sooner!
I did my best! I have a workshop I’m setting up too.
Fine, that’s fine. I’m sure you’re doing your best… But I still need to do what I need to do!
Jack, on being rescued
What starts it, and why the battle appears when it does
Workshop Hours is the quest before this one. It asks you to fill a town order and then talk to Jack at the Town Warehouse, and the order is one 
Supply: Iron crate, which is the same supplies system the rest of town orders runs on.
Handing that over does three things at once. It starts this quest, it starts Plaza Battle, and it raises Jack a vendor tier. That is why the Market Square icon appears on the map at the same moment the workshop quest does, and why the two feel like one job.
I’ll handle the production plans at my workshop. But the easy stuff? Well that’s on you.
The easy stuff?
Yeah… You’ll just need to clear out all those undead barbarians.
That’s sarcasm. Right?
Jack, handing over the easy stuff
| Docked and Found | Find Jack in the liberated part of the port | 10 reputation, and 10 town quality |
| Workshop Hours | Fill a Supply: Iron order, then talk to him at the warehouse | 10 reputation, 1 Town Gratitude, and his shop goes up a tier |
| Jack’s Back in Business | Win the Market Square Purge, then talk to him at the warehouse | 10 reputation, 2 Town Gratitude, and a Story |
| Give materials | 5 Building Kit I and 5 Wooden Kit I, handed to him | The workshop rebuilt, and you are his apprentice |
The list he hands you next
Closing this one opens two more, and they run in parallel rather than in order. 
Give materials wants 5 
Building Kit I and 5 
Wooden Kit I, given to him directly, and whichever you hand over second finishes the job.
He also points you somewhere, and it is the first the game says about the room under your own base.
Be creative, my friend… I’ve heard rumors of an Inquisitor workshop buried somewhere under your base.
If even half the stories are true, you might find exactly what you need down there.
Jack, on where to make his kits
The reward for finishing it is not the kits back. He rebuilds his workshop, signs the town campaign he had been refusing, and takes you on as his apprentice, which is the point at which his reputation starts mattering more than anything he sells.
Why his reputation is worth more than the quest
Every quest in this chain pays 10 reputation with Jack, and the technology tree has three nodes that cost no spheres at all and open on that number instead. They sit at 40, 50 and 70.
The one at 50 is the expensive one to be short of. It sits under Conveyors, which means it sits under the Factory, every assembly bench, and eight of the ten shops the town can build, including 
the Smithy, 
the Alchemist and 
the Jeweler. A run that skips Jack’s quests stalls there without the tree ever saying why.
| Building branch | 40 reputation | Tool Rack II and the line into Conveyors |
| Building branch | 50 reputation | Conveyors, the assembly benches, the Factory, and every shop from Artisans I onward |
| Metallurgy branch | 70 reputation | The far end of the metal line |
Questions about Jack’s Back in Business
Why is Clear the zombies around his workshop not ticking off?
Because the quest has no completion check of its own. It closes when you talk to Jack, and he is at the Town Warehouse rather than at the workshop. Winning the battle is only half of it.
Which battle clears the zombies around the workshop?
The Market Square Purge, and only that one. It is a separate journal entry called Plaza Battle, and the Eastern Districts purges, the sewer brawls and the floodgate battles do nothing for this quest.
Where is the Market Square battle?
Under the Square label on the map, north of the crossroads where the guards camp outside the gate. Head up past the nun and it is the icon beyond the wall.
Why will the fight not start?
It wants 22 defense power and two squads, which is double what the sewer brawls ask for. Barricades and towers are where that number comes from.
What do you get for finishing it?
10 reputation with Jack, 2 Town Gratitude and a Story, plus the Give materials quests that rebuild his workshop for 5 Building Kit I and 5 Wooden Kit I.
Is Jack worth the reputation?
Yes, more than most. Three technology nodes cost no spheres and open on his reputation instead, at 40, 50 and 70, and the one at 50 gates Conveyors, the Factory and eight of the ten shops the town can build.
Leads To
Rewards
Comes After
More Quests
A Cure for Wellness
A Good Example is the Best Sermon
Any Portal in a Storm
Battles in the Sewers
Book Broker
Brawns and Brains
