How did Duras find out Kurn was Worf's brother?

Question

I've always wondered how he managed to find that out. As I just saw the scene where he fights Kurn in "Sins of the Father."

Answer 1

It's not stated explicitly on-screen, but it doesn't take much to connect the dots, as his identity was not hidden very well.

Here's how it played out:

Klingon warriors capture a Romulan ship, learn of a Klingon traitor at Khitomer, but not who it is.
The whole Klingon council knows it has been Ja'rod, father of Duras, but Duras is powerful - so they decide to blame it on Mogh. From their perspective, the only one left of that family is weak and not even a real Klingon. They did not expect Worf to challenge the judgement, so this seems to be an easy way out:

K'MPEC: His family is powerful. If the truth were known, it would shatter the Council, most certainly plunge us into civil war. You were in Starfleet. We did not expect you to challenge the judgment, nor did we know there was another son of Mogh.

(TNG: Sins of the Father)

Now they set their plot into motion, but unexpectedly, some Commander from some minor house with no direct contact to Worf before stands up and even becomes his cha'DIch. What the hell? He introduces himself as:

KURN: I am Kurn, son of Lorgh. I will stand by Worf's side. I am cha'DIch.

(TNG: Sins of the Father)

From the perspective of the Council, this was a complete surprise. They needed to find out the motivation of Worf and that Kurn guy. K'mpec tries to find out Worf's motivation on screen:

K'MPEC: You should not have brought the challenge. There was no risk to you. What does it matter?
WORF: I am Klingon.
K'MPEC: Of that I have no question. But your life in the Federation would not be affected by this judgement.
WORF: My father...
K'MPEC: Is dead. He died long ago. I knew your father, served with him. This is not how I wanted to remember him. We must let the past be and protect what we have now. If you leave before the Mek'ba, no shame will come on you. Return to your ship. Go back to your life. The challenge will be forgotten.

(TNG: Sins of the Father)

It's not a stretch to assume that off-screen, people have been trying to figure out why Commander Kurn would be involved. And the answer is a combination of the following facts:

• Mogh had a son named Kurn who is thought to have died at Khitomer
• Shortly after, Lorgh, a friend of Mogh's, finally has a son - coincidally also called Kurn

KURN: I was barely a year old when you left for the Khitomer Outpost. You, our mother and father were not going to stay long. It was decided that I did not need to go. I was left to stay with our father's friend, Lorgh, until you returned. You never did.
WORF: The Starfleet officer that rescued me was told by the Klingon High Command that I had no living relatives.
KURN: They assumed that I was killed with the family at Khitomer. Lorgh had no sons. He took me into his family. It was not until I had reached the Age of Ascension that I was told the truth.

(TNG: Sins of the Father)