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Shakuni, son of Subala, was king of Gandhara.

In Mahabharata, who killed the Shakuni during the Kurukshetra war?

Sahadeva has killed Shakuni in the last day of the battle. During the dice game, he vows to kill Shakuni for his deceit.
From

Book 9: Shalya Parva Section 28

Shakuni took up a formidable mace and hurled it at Sahadeva. That mace also, unable to achieve its object, fell down on the Earth. After this, Subala's son, filled with rage, hurled at the son of Pandu an awful dart that resembled an impending death night. With the greatest ease Sahadeva, in that encounter, cut off, with his gold-decked shafts, into three fragments, that dart as it coursed swiftly towards him. Cut off into fragments, that dart adorned with gold fell down on the earth like a blazing thunderbolt from the firmament, diverging into many flashes. Beholding that dart baffled and Subala's son afflicted with fear, all thy troops fled away in fright. Subala's son himself joined them. The Pandavas then, eager for victory, uttered loud shouts. As regards the Dhartarashtras, almost all of them turned away from the fight. Seeing them so cheerless, the valiant son of Madri, with many 1,000 shafts, checked them in that battle. Then Sahadeva came upon Subala's son as the latter, who was still expectant of victory, was flying away, protected by the excellent cavalry of the Gandharas. Recollecting, O king, that Shakuni, who had fallen to his share, was still alive, Sahadeva, on his car adorned with gold, pursued that warrior. Stringing his formidable bow and drawing it with great force, Sahadeva, filled with rage, pursued the son of Subala and vigorously struck him with many shafts equipped with vulturine feathers and whetted on stone, even like a person striking a mighty elephant with pointed lances. Endued with great energy of mind, Sahadeva, having afflicted his foe thus, addressed him, as if for calling back to mind (his past misdeeds), in these words, 'Adhering to the duties of a Kshatriya, fight (with me) and be a man! Thou hadst, O fool, rejoiced greatly in the midst of the assembly, while gambling with dice! Receive now, O thou of wicked understanding, the fruit of that act! All those wicked-souled ones that had ridiculed us then have perished! Only that wretch of his race, Duryodhana, is still alive, and thyself, his maternal uncle! Today I shall slay thee, striking off thy head with a razor-headed arrow like a person plucking a fruit from a tree with a stick!" Saying these words, O monarch, Sahadeva of great strength, that tiger among men, filled with rage, rushed impetuously against Shakuni. Approaching his enemy, the invincible Sahadeva, that foremost of warriors, forcibly drawing his bow and as if burning his foe with wrath, pierced Shakuni with ten arrows and his steeds with four. Then cutting off his umbrella and standard and bow, he roared like a lion. His standard and bow and umbrella thus cut off by Sahadeva, Subala's son was pierced with many arrows in all his vital limbs. Once again, O monarch, the valiant Sahadeva sped at Shakuni an irresistible shower of arrows. Filled with rage, the son of Subala then, single-handed, rushed with speed against Sahadeva in that encounter, desirous of slaying the latter with a lance adorned with gold. The son of Madri, however, with three broad-headed arrows, simultaneously cut off, without losing a moment, that uplifted lance as also the two well-rounded arms of his enemy at the van of battle, and then uttered a loud roar. Endued with great activity, the heroic Sahadeva then, with a broad-headed arrow, made of hard iron, equipped with wings of gold, capable of penetrating every armour, and sped with great force and care, cut off from his trunk his enemy's head. Deprived of his head by the son of Pandu with that gold-decked arrow of great sharpness and splendour like the sun's, Subala's son fell down on the earth in that battle. Indeed, the son of Pandu, filled with rage, struck off that head which was the root of the evil policy of the Kurus, with that impetuous shaft winged with gold and whetted on stone. Beholding Shakuni lying headless on the ground and all his limbs drenched with gore, thy warriors, rendered powerless with fear, fled away on all sides with weapons in their hands. At that time, thy sons, with cars, elephants, horse and foot entirely broken, heard the twang of Gandiva and fled away with colourless faces, afflicted with fear and deprived of their senses. Having thrown down Shakuni from his car, the Pandavas, O Bharata, became filled with delight. Rejoicing with Keshava among them, they blew their conchs in that battle, gladdening their troops. All of them, with glad hearts, worshipped Sahadeva, and said, "By good luck, O hero, Shakuni of wicked soul, that man of evil course, hath, with his son, been slain by thee!'"