This is one of the points at which an unpleasant surprise starts waiting for you at the Shrine of Talos, so if you want to get a few more things done before then, you’ll have to stop returning there for now.
When it’s time to head for the Warrens, you’ll want to seek the lowest parts of Markarth. Step over to the smelter and you may well find Mulush Gro-Shugurz, an orc who is in charge of the smelter. While he’s not terribly friendly, he isn’t inclined to be grumpy either and talks fairly easily. According to him, they lose workers pretty frequently: most of them declare that they’ll join the Forsworn and run off into the hills. This is pure foolishness in his eyes, as running off to dedicate your life to killing the Nords is pretty meaningless to the practical Orsimer.
You can ask him about the attack, and he’ll complain about Weylin, who apparently left work to go commit suicide by proxy before the smelter could meet the silver quota, leaving the Orc behind on his work. He claims not to know anything else of interest, but a successful attempt at coercion, bribery, or intimidation will get out of him that Weylin recieved a scrap of paper alongside his most recent paycheck, and scuttled off to his room in the Warrens with it immediately.
Mulush, however, doesn’t have the key- that would be in the possession of Garvey, a man for whom Mulush spares nor loses no love at all. And Garvey, thankfully, can be found in the Warrens itself. Therefore, turn from the smelter and cross the wood bridge to the Dwemer metal door at the lowest point in Markarth so that you can head into the Warrens.
Garvey, thankfully, is waiting just inside the Warrens, off to the right. He’s leaning on the wall there, and can provide you not only with information, but also Sneak training- and Weylin’s key.
According to Garvey, Weylin was known to him- but he won’t share any further information. When you attempt to get Weylin’s key out of him, you’ll have to apply persuasiveness, terror, or a bit of money to get what you seek- but you’ll get it if you’re lucky. If you’re unlucky, you may have to use a lockpick- but the locks down here are easy, and the area is well-shadowed, so it’s really simple to get into any of the quarters.