Giving equipment to Malborn:
When you say you are ready, an inventory screen will appear. Navigate to the item(s) you want. On the bottom left corner it will tell you what button to press for "Give".
It is possible to do the whole thing sneakily, but you won't be able to save Malborn and the prisoner. So you don't need to give Malborn anything (i.e. don't stress if you didn;t manage this part and are basically naked in there). But if you want to save them, you will need to fight two people at once so you may as well give him everything (except some clothes to wear) including lots and lots of potions, as you won't find many in the embassy.
At the party:
Get a drink from somebody (the serving girl or Malborn) and give it to the raving drunk sitting to your right as you enter. He will make a scene so you can slip away. Go to Malborn behind the bar and follow him. Keep following him and the chest where your stuff is will become a quest marker so you can find it.
In the embassy:
Go through the other door and you are in a hallway. Sneak near the open doorway and listen to the guards' conversation. Eventually they will walk to the other end of the room. You can sneak past them to the right where there are some stairs.
Be careful going through the hallway up here, there is a hidden mage who will spot you and start a fight. You can fight him without anybody else coming to his aid, or you can carefully sneak past him.
There will be a room (I think the quest marker points to it) in which there is a guy sitting at a desk with a chest behind him. Sneak behind him and take the contents of the chest. I didn't fight him so I don't know if that will arouse suspicions or not.
The dungeon:
Read one of the letters which will talk about a prisoner in the dungeon. There are some stairs leading down nearby. That leads to a door to the dungeon.
While the guard is torturing the poor guy, sneak behind the guy at the desk and take the contents of the chest. Then sneak to the shadows near the hay pile/other stairs upward.
If you intend to save the prisoner and Malborn, get a sneak shot in or two then kill them both. If you intend to save the prisoner, don't let him go just yet or you may have a hard time keeping the next guys away from him. If you intend to take the sneaking route, make your way to the hay pile, right next to the other stairs, and be in sneaking mode.
Then, very important:
READ THE BOOKS. ALL OF THEM!
This causes the "Search for info" quest to be completed and the "Escape the embassy" one to start.
Almost immediately, you will hear two guards come in by the door that you originally came in. They will be talking to you, ("Come out, spy, you're surrounded!" etc) and occasionally beating up Malborn.
If you intend to save the prisoner and Malborn, you can kill them now. Then take the key from their body, let the prisoner go, and escape down the trap door (there is a quest marker pointing to it).
Otherwise, stay in your position near the hay pile. Eventually one of the guards will go up the stairs, presumably looking for you, while the other guard is busy doing something near the desk. When they go up, pickpocket the key off them. You might want to save just before you try in case you're not successful the first time. While one guard is upstairs and the other is busy at the desk, sneak to the trapdoor and you're free!!!
In the tunnel:
There is a big scary troll. Best to run out as fast as possible. If he sees you, he doesnt come out right away (or at all?) and you can fast travel the heck out of there.
Also, there is a section you can walk backwards up, and the troll can't reach you. Excellent place to practice ranged spells/archery, since the troll regenerates health. You have to continually walk backwards though or else you slip off.
