What "mighty works" was Sodom deprived of that would have prevented its destruction?
It might be helpful to look at another translation of this, as well as the surrounding passages:
Matthew 11:20-24, NASB: "Then He began to denounce the cities in which most of His miracles were done, because they did not repent. 21Woe to you, Chorazin! Woe to you, Bethsaida! For if the miracles had occurred in Tyre and Sidon which occurred in you, they would have repented long ago in sackcloth and ashes. 22Nevertheless I say to you, it will be more tolerable for Tyre and Sidon in the day of judgment than for you.
23And you, Capernaum, will not be exalted to heaven, will you? You will descend to Hades; for if the miracles had occurred in Sodom which occurred in you, it would have remained to this day. 24Nevertheless I say to you that it will be more tolerable for the land of Sodom in the day of judgment, than for you." (emphasis added).
Christ is denouncing the cities that have ignored His miracles — His "mighty works". These cities include Chorazin, Bethsaida, and Capernaum. He is telling His audience that the cities of Tyre, Sidon, and even Sodom would have long since repented at the sight of the miraculous signs He has performed.
Jesus is amazed at the spiritual blindness of Chorazin, Bethsaida, and Capernaum to all of His fantastic works while others from the past would have repented far sooner given the same opportunity. We may not know just what signs God gave to the distant cities in question (although we may), but Christ is simply stating that the signs He performed in these cities would have been enough to at least convince Tyre, Sidon, and Sodom to turn from their ways. We might understand that, while many miracles occurred in ancient times, the purpose of Christ's miracles was to explicitly demonstrate His divine authority over nature throughout His ministry.
We might look to the ancient city of Nineveh (as per the OP) as an example of this:
Jonah 3:4: "4Then Jonah began to go through the city one day’s walk; and he cried out and said, 'Yet forty days and Nineveh will be overthrown.”
These 8 simple words were enough to convince the people of Nineveh, beginning with the king himself, that they immediately had to fast in sackcloth and ashes: from royalty all the way down to the lowliest peasant. Beginning in chapter 3, the king even issued a proclamation:
Jonah 3:7b-8: "Do not let man, beast, herd, or flock taste a thing. Do not let them eat or drink water. 8But both man and beast must be covered with sackcloth; and let men call on God earnestly that each may turn from his wicked way and from the violence which is in his hands."
This is an incredible reaction by a Gentile nation, one steeped in wickedness, that believed so few words of warning by Jonah and acted on that prophecy to save themselves so that:
Jonah 3:9: "God may turn and relent and withdraw His burning anger so that we will not perish.”
By doing these things in a spirit of great humility, the ancient city of Nineveh was spared from God's anger — unlike the cities mentioned above that rejected far more powerful evidence of Christ's other-worldly authority.