Chiang Kai-shek didn't want to lead a unified KMT. In fact, he totally planned, executed, and then perpetuated the persecution of communists and the CCP from 1926-1937. And to be fair, Sun Yat-sen also had no interest in seeing the party unified with communists either, he was forced to by the USSR who was funding the KMT's Northern Expedition.

The KMT's creation led it to attract a plethora of plural-minded individuals. The three major groups can be broken down into "Liberals:" Those who sought to create a sort of modern capitalist-federation in the likes of the US; "Communists:" by 1926, no one had universally agreed to what communism was or meant. They just knew that it had to make China a better, modernized state like it did for Russia in the USSR; and finally "Nationalists:" This group is kind of nebulous. Some in recent years have likened them to being fascist minded, while others note that these men adopted their own Chinese concepts of modern military totalitarianism. So Liberals, Communists, and Nationalists, and they're all working "together" under a singular party, the KMT. The Chinese didn't really have the whole 'political parties' thing nailed down the way it is in federal state like the US. But they were all unified in one major goal: Unite China, Modernize China, and Make China Great Again by any means possible!

Solidifying Chiang's Power; the Canton Coup and the Northern Expedition

When Sun died, there was no real successor for the KMT. Chiang and Sun, although friends, had a very tense political relationship due to Chiang's refusal to cooperate with the CCP. Upon Sun's death, Chiang had an edge over other generals; he was the head of the Whampoa Academy, which meant that he not only had a professional corps of loyal officers, but an army at his disposal located right in Canton. But he himself really had no legitimate means of being the leader of the KMT in the eyes of every other KMT general. They were all the same; a group of former cadets trained in the art of military at the Imperial Japan Military Academy that were unified by their youthful bravado to return to a fragmented homeland and not just unify it, but also turn it into a great world power that could stand up to the Westerners. All of them would play a key role in modern Chinese history, and most of them wanted the leadership role, along with various politicians.

Chiang needed to act fast if he was to keep his position as head of KMT, or even his head. Chiang chose a combination pretty common to most dictators, assassinating/arresting those who could threaten him, and then accelerating war preparation to give other generals something to be busy with other than plotting his demise. One of the first things he did upon the death of Sun was killing Liao Zhongkai and arresting Hu Hanmin, getting rid of two potential rivals. Next, Chiang instigated the "Canton Coup" (also known as the Zhongshan Incident), where he used his power as head of the Whampoa Military Academy to attack certain communists, arresting Zhou Enlai and exiling Wang Jingwei, but also in the process forcing Soviet advisers to flee. Despite the close calls and tensions, Stalin decided to continue his backing of Chiang and the CCP stayed as an artery of the KMT, though not for long.

After ensuring he had control over the communists, Chiang then had to deal with fellow military generals. They weren't communists and they all weren't unified in who should replace Chiang, but many were convinced that Chiang should be disposed, a feeling that would soon dissipate as Japanese aggression became more and more belligerent. In order to cull the generals, Chiang basically took a gamble; he would launch the Northern Expedition in 1926 rather than later, against the advice of the Soviet advisers the year prior. Today we know the NE was a stunning success in many ways for the KMT, although it ultimately fell short. But at the time there was no guarantee the campaign would succeed. Fortunately for Chiang, as the KMT Army marched further north it became apparent that certain warlords had no real allegiance to their leaders, and Chiang's willingness to use what he called "Silver Bullets" (Bribes) meant that most of the warlords fell apart quite quickly; some even proved to be loyal allies in the future (Tang Shengzhi). But by 1928, Chiang was the undisputed leader of the KMT due to a combination of a successful war solidifying his leadership and him assassinating or exiling potential rivals.