To understand the prevalence of the ii-V-I in jazz, there are three good places to look: ragtime, Tin Pan Alley, and blues.

###Ragtime

Ragtime was characterized by something called the "Ragtime Progression":

| (V7/V/V/V) | V7/V/V | V7/V | V7 | I |

or more simply:

| III7 | VI7 | II7 | V7 | I |

In the key of C major, this progression is:

| E7 | A7 | D7 | G7 | C |

The first line illustrates the "centripetal" movement around the circle of fourths. In other words, each chord serves a dominant (V) function which leads to the next V chord. But even beyond this, the ragtime progression was interesting to people like Scott Joplin because of the possibility it opened for chromatic movement in the voice-leading, in the melodies, etc. Ragtime composers were so interested in exploring those possibilities that the III⁷-VI⁷-II⁷-V⁷-I progression became a hallmark of ragtime.

###Tin Pan Alley

Then in the early 1900s, songwriters like Cole Porter, Irving Berlin, and George Gershwin came onto the scene. Their music was often referred to as Tin Pan Alley, and their songs were so influential that they form something called the Great American Songbook. These songs are still played with impressive frequency today in jazz. As it turns out, these composers were heavily influenced by ragtime:

The first generation of Tin Pan Alley composers was obsessed with ragtime and its musical and commercial possibilities. They were also interested in repackaging the instrumental ragtime compositions that Scott Joplin and his colleagues had developed as the foundation for their own novelty songs. The decade of Tin Pan Alley's rise, the 1910s, might be usefully marked off by the debut of Irving Berlin's “Alexander Ragtime Band” in 1911 on one end, and George and Ira Gershwin's “The Real American Folk Song (Is a Rag)” in 1918. (p. 3 of http://www.lfpl.org/mylibraryu/pdf/Session_Two_Materials.pdf)

Given the massive influence of these Tin Pan Alley composers and the endurance of their songs, the II-V-I progression would forever become an integral part of jazz harmony. But in part because Tin Pan Alley composers often wrote for musicals, their music followed specific forms like AABA, AAB, AB, etc. which each letter (A and B) marking different sections. For variety, these composers used different sections to modulate to different keys. This had two effects: (1) the prevalence of the ii-V-I grew because it's perhaps the easiest/most natural way to modulate, and (2) the ii-V-I became preferred over the II-V-I, partly because the ii-V-I was entirely diatonic and helped establish the new tonal center more quickly.

###Blues (and Bebop)

There's a second unique lineage that also explains the prevalence of the ii-V-I. The core 12-bar structure that evolved from blues is something like this (in C):

| C7 | C7 | C7 | C7 |

| F7 | F7 | C7 | C7 |

| G7 | F7 | C7 | C7 |

As jazz musicians began using the blues progression, many sought ways to increase its complexity while preserving its original structure. This was especially true of the bebop musicians. People like Charlie Parker were known for adding chords into a song as a way to enable more complex melodies and improvisations. The easiest ways to add in extra chords are: (1) add the V⁷ chord in front of any "orphaned" I chords, and (2) add the ii chord in front of any "orphaned" V⁷ chords. The result is an explosion of ii-V and ii-V-I progressions in bebop music. For example, many bebop musicians have taken the blues structure above and modified it this way:

| C7 | Bø E7alt | Amin D7 | Gmin C7 |

| F7 | F7 | C7 | Emin A7 |

| Dmin | G7 | C7 A7 | Dmin G7 |

All we've done here is add additional V⁷ chords and ii chords. This procedure of adding ii-V⁷ chords quickly spilled over into many other songs that didn't have the blues structure. And this simply increased the prevalence of the ii-V-I even more.