Brainpower? Everyone just seems to be typing the names of Greek Islands.
Ok, so from previous posts it not Crete, Lesbos(lesvos), Rhodes, Limnos, Samos, Kos, Corfu, Zakynthos/Zanti, Naxo, Amorgos, andros, Felegandros, Khios, Skiathos, Thassos.
From your clue, mykonos. Keas capital is nothing like Kea, same with Milos. That ruled out a few more. Prob not Ios as Capital is called Nora. Not Kythnos as capital is called Hora.
Is it Astypalea? Also know as Stampalia or Astropalia. Capital Astypalia?