I still can't complete this quest even after downloading all the updates. I get to the point where Jorlief tells me to show the necklace to Calixto, and I can sell him the necklace and then the quest just stops and nothing happens and I can't talk to Jorlief Calixto Viola or the mage about it anymore. The only thing I can do is steal the necklace back from Calixto, and then I can repeat the process and the guards or Jorlief tell me to go to Calixto and nothing happens again after that.

From what I can tell there's an important step you need to take so that you can speak to Viola. I had to read the UESP to find out what I needed to do and it appears you have to take one of the flyers regarding "the butcher" from the manor house in Windhelm while you're doing your investigation there. If you have the flyer on you, then you can talk to Viola and that should put you on course to the next part of the quest. It is a bit obscure and the game does nothing to let you know what you need to do, but as far as I can see, that's it. Hope this helps.

Did you get the journals from Hjerim?

If you sold the amulet to Calixto you can retrieve it later off his dead body.

I never had a problem with this quest until recently but I was able to move it along by doing the following. I don't know if it will help you though.

I was able to finish the first half with the one person arrested and I had sold the amulet to Calixto. I left Windhelm and stayed away for a while so the second part would trigger. It never did when I went back a few different times. There was never another murder. I ended up just going to the prison and speaking with the prisoner and it gave me the appropriate dialogue to tell him there was another murder, (even though I never saw it and I know where it was supposed to be since I'd done this quest a few times before)...and then it continued on as normal. I hope something in that helps someone get through it.