Why did Sodom and Gomorrah perish?

One of the best-know stories from Scripture is that of Sodom and Gomorrah, where two cities were reduced a smoldering rubble by fire from heaven, and Lot's wife was turned to a pillar of salt after she looked back as her family was fleeing the city. The reason for their destruction was pretty simple - the depth and range of their sin against God had cried out to heavens, and God sent angels to rescue Lot and his family before they destroyed it (Gen 19:13).

The puzzle, though, comes from this citation:

And you, ^Capernaum, will you be exalted to heaven? You will be brought down to ^Hades. For if the mighty works done in you had been done in Sodom, it would have remained until this day. ^But I tell you that ^it will be more tolerable on ^the day of judgment for the land of Sodom than for you.” (Matt 11:23-24)

In other words, Sodom could've been saved if the works that had been done in Capernaum had been done in Sodom. But why didn't God do those works in Sodom's time? That's a to ask on the other side of eternity, but the following passage provides a clue:

"...if by ^turning the cities of Sodom and Gomorrah to ashes he condemned them to extinction, ^making them an example of ^what is going to happen to the ungodly..." (2 Peter 2:6, see also Jude 7)

Sodom, Gomorrah, and the entire valley they inhabited were overthrown because of their sin, but the manner in which they perished demonstrated to all the following generations of what will happen to those who die in their sins outside of faith in Christ.

All citations from The Holy Bible : English standard version. 2001 Wheaton: Standard Bible Society.