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.
If you are being attacked you may have some sort of bounty. You can either yield (sheath your weapon) or if they are currently attacking you on site you may need to leave and come back a few days later.
The following answer to this may be considered a spoiler so:
Spoiler Alert:
What you should have gotten was a note to rent a room in the
sleeping giant inn. Once you have rented the room and gone to it,
Delphine gives you the horn and then asks you to follow her. She goes
to a room directly across from the room you rented and through a
secret passage. The horn is actually returned to High Hrothgar,
Arngeir is expecting it. Delphine however wants you to accompany her
to another location to prove you are the dragonborn.
http://www.uesp.net/wiki/Skyrim:The_Horn_of_Jurgen_Windcaller
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