The fascinating thing about the two Lankhmar adventures, Swords of the Undercity and Swords of Deceit, aside from the implication that all Lankhmar stories revolve around swords, is that they wind up telling me more about the city than the book specifically dedicated to the task. Not in terms of quantity of pure information, to be sure, but definitely in the sense that they convey the feeling of living in Lankhmar much better than the campaign book.
Probably because there aren't quite so many lists. There are some. I'm confident that I could tell you about all the furniture in the vampire Myrria's townhouse (although in an unexpected departure from the old school module format, most of the rooms have neither monsters nor treasure, making the repeated listing of furniture seem almost pointless). However, that list of rooms is actually only one small part of a more interesting story - Myrria is a Lankhmar vampire, which means she has an uncontrollable bloodlust every full moon. During one of these rampages she killed a priest of the Gods of Lankhmar (those italics are obligatory, which I think comes across as nicely sinister), and now, for the first time in years, the Night of Fear, in which the Gods of Lankhmar stalk the streets of the city, is occurring on a full moon. Myrria first shows up playing the ingenue, as part of a convoluted plan to get the PCs to fight the assassin demon for her. Then she goes into a blood rage and you kill her and take her furniture.
And that's only one of the three stories in Swords of Deceit. The others are a creepy revenge story where a noble buries his older brother alive in order to inherit the title, but the brother escapes the grave, runs away to a foreign land to study magic for 20 years, and then comes back and starts tormenting his betrayer. I feel like a more modern version of the story would be more sympathetic towards the brother instead of just portraying him as an evil wizard who must be stopped, and the end of the story relies on the PCs being unable to save the noble's life. But it's a good framework for a game. If I were running it, I'd probably stretch it out to a mini-campaign of two or three adventures and give the PCs more agency in how they want it to resolve.
The other adventure in Swords of Deceit is a sequel to what I'm assuming is the most off-the-wall of the Lankhmar books, The Swords of Lankhmar. To recap the recap, what happens is this - intelligent rats try to take over Lankhmar, the Gods of Lankhmar try to stop them, but they're defeated when the heroes manage to summon the War Cats to battle. A lot of other weird stuff happens, including the Gray Mouser drinking a shrink potion and infiltrating the rat society, but I'm going to be honest - I'm surprised to see the plot return. I figured if they were going to base an adventure around one of the Lankhmar books, they'd have chosen one that wasn't goofy as hell.
Still, the disgraced former King of the Rats is back and out for revenge. Worshiping a new and foreign Rat God, he and his fanatical followers have kidnapped Fafhrd and the Gray Mouser, potentially throwing the human-rat treaty into peril. The PCs must drink Lankhmar's disgusting shrink potion (it conserves mass, leaving a puddle of organic goo behind that you must rejoin before the potion wears off) in order to visit the city of the rats, using a combination of stealth, diplomacy, and combat to rescue the canon characters. It's a wild, wild story.
Swords of the Undercity is the less ambitious of the two, though still a pretty interesting story. There's a gem that turns people into sewer mutants and the sewer mutants want it back, because it's the only way they can reproduce. It's not clear why they can't be allowed to have it, but it may be because the PCs were the ones who found it in an old temple. That's the rule in rpgs - if you dig it up, you get to sell it. Or maybe it's because they kidnap people and turn them into monsters against their will. Despite being a straightforward fantasy tale, it justifies being set in Lankhmar - a corrupt noble tries to double cross you when he learns the fake treasure map he sold you was real, the middle act is all about how fencing valuable loot is fraught in a city with a thieves' guild, the sewer mutants hire an assassin.
It's funny. After reading Lankhmar: City of Adventure, I felt like maybe the setting was overhyped. People liked the books, so they gave too much credit to a licensed rpg that was often kind of bland. However, after reading these adventures, I get it. This is the sleazy, out of control "dungeon crawl meets crime caper" sort of game that I sensed in potentia. It was real, and not just something people projected onto a sub-par book out of wishful thinking. However, by the same token, I don't think you can just use the main Lankhmar book on its own. These adventures show us a living city, suitable for PCs as wild as Fafhrd and the Gray Mouser themselves. Honestly, they're my favorite AD&D modules yet, and probably the only ones besides Reverse Dungeon that I would even consider running.
Ukss Contribution: From Swords of the Undercity, the fence who double-crosses you has a lot of interesting furniture (these books are much better than earlier AD&D modules about the furniture-to-story ratio, but there's still a ton of furniture), but because his main profession is "religious extortionist" (basically, he acquires religious relics and ransoms them back to their temples), his furnishings consist of many strange and exotic wonders - like a bathtub made from the half-shell of a giant clam. It was given to him by the priests of an aquatic god and it's a simply wild bit of characterization. Somebody big and flamboyant in Ukss is going to get something similar.
In Swords of Deceit, the rat city has a location called "Cat's Cradle," named for a legendary rat leader who kept a cat as a pet. Maybe one of the intelligent rats of Ukss can also be so bold (what can I say, I love it when my NPCs live large.)