You want to get outside into the courtyard. I'm not sure exactly where you are, but if you are where I think you are, you should come across a hallway before going upstairs. You should find an exit there that leads you directly to the courtyard, avoiding the need to go upstairs.
Once outside, you need to get to the garden (just follow the wall of the courtyard: watch out for guards) and the solarium.
Once inside, you'll hear Rulindil talking to his subordinate. Once he leaves, head to the room to the right of where Rulindil enters (again, watch out for guards). Here, ransack the room and look for dossiers found inside of a chest. You'll also find a interrogation chamber key.
Once you have that, you need to escape. To do that, head down into the dungeon, once more dealing with the guards that you find there. You'll find a prisoner locked up as well as a chest that contains another dossier.
Finally, you'll need to speak to and release the prisoner: Etienne Rarnis. Once freed, you'll be rushed by even more guards. They'll have the trap door key, which you can then use to escape the embassy directly from the dungeons.
When you drop through the trap door, you'll enter a cave where you'll have to fight a troll.
After that, you're home free.
From the UESP wiki's "Blood on the Ice - Bugs" article:
Wuunferth Arrested Variation
If you speak to Wuunferth directly, instead of Jorleif, then Calixto
will attempt to murder Arivanya in the Stone Quarter. If you choose to
have Wuunferth arrested, then Arivanya will be found murdered. For the
final showdown, a random, female NPC will be chosen as the market
victim. If the game runs out of potential victims, one of two bugs
will trigger:
- After speaking to Wuunferth, the journal will immediately update to stage 130. After chasing Calixto into Hjerim, he cannot be killed. He
then will not leave Hjerim and becomes hostile when you enter.
- Calixto will stay inside his House of Curiosities and will never attempt his last kill.
- If you buy Hjerim and arrest Wuunferth, when you get the objective to patrol Windhelm's streets Calixto may appear inside Hjerim, in the
room behind the wardrobe, and use any equipment you may have left in
it. He will be immediately aggressive and killing him advances the
quest normally.
I can hit him until he falls to his knees but he gets back up and carries on.
Updating Skyrim should fix this bug. In March 2012 (PC) and April 2012 (consoles), update patch version 1.5 was released. One of its bug fixes is:
Fixed issue where Calixto would fail to die properly in Blood on the Ice.
If playing on the PC, make sure to install the Unofficial Skyrim Patch (USKP) as it fixes the bugs left unfixed in the 'Blood on the Ice' quest by the official patches. List of bug fixes by USKP related to the Blood on the Ice quest here.
Best Answer
"Diplomatic Immunity" is part of the Main Quest Line. You must first do the main quest line first. After you do the quest "Horn of Jurgen Windcaller", talk to Delphine in Riverwood and say you want an attic room. Go to the room and wait for her to come and talk to you and a quest should start. After completing "A Blade in the Dark", "Diplomatic Immunity" should start.