Mongols conquer surviving Komnenid empire and create another "Golden Horde". What does it look like?

[Alligator]

Scenario:

• The Komnenid empire survives briefly longer, with Manuel winning Myriokephalon and Alexios II succeeding at a later age. There is no Fourth Crusade, so the Byzantines are worth conquering.
• The Mongol conquests proceed as historical, except the Byzantines are in one piece.
• Even though the Byzantines could probably pay off or fight off the Mongols, they are defeated soundly and Constantinople is taken.
• A "Golden Horde" or "Ilkhanate" like entity is created in Greece and Anatolia, called the Purple Horde.

How does this new khaganate develop? What are its key cities, like the Golden Horde's Sarai Batu and Astrakhan? What religion does its khagan convert to? How are Byzantine and Mongol traditions reconciled into a new imperial administration?

 

[BBadolato]

Scenario:

• The Komnenid empire survives briefly longer, with Manuel winning Myriokephalon and Alexios II succeeding at a later age. There is no Fourth Crusade, so the Byzantines are worth conquering.
• The Mongol conquests proceed as historical, except the Byzantines are in one piece.
• Even though the Byzantines could probably pay off or fight off the Mongols, they are defeated soundly and Constantinople is taken.
• A "Golden Horde" or "Ilkhanate" like entity is created in Greece and Anatolia, called the Purple Horde.

How does this new khaganate develop? What are its key cities, like the Golden Horde's Sarai Batu and Astrakhan? What religion does its khagan convert to? How are Byzantine and Mongol traditions reconciled into a new imperial administration?

I don't think a Golden horde analogy would work it would be more IlKhanate. The Golden Horde suffered near constant succession disputes, you would be lucky to have a khan reigning more than 10 years.

Realistically I could see this Purple Horde not Constantinople but at least take Anatolia so I would say more second Sultanate of Rum with Orthodox Greek and Armenian along with Muslim Turks as subjects. Key cities might just be key cities in Anatolia.