In the long run, Chiang Kai-shek and the KMT are able to consolidate power.
In the short run, early elimination of the Communists probably produces more pressue on Chiang to defy the Japanese. However, there will be no Xi'an Incident which forced Chiang to make promises to defy the Japanese (which he kept). So he might be able to delay the start of the Sino-Japanese War which would only benefit him. There would be a lot of popular unrest against Chiang because of that, but he should have enough power to withstand it. Very hard to say. Everything depends on how you think things will develop. Small decisions could have big impacts.
Without the Communist threat, Chiang won't need to keep his best troops near Yenan to keep an eye on the Communists. Instead, they could be used to defend other areas. This might change the battlelines in China to be slightly more favorable to China which reduces some of the strain the KMT economy suffered from. If the KMT is able to hold onto Canton and Wuhan, it is in a much better strategic and economic position.
It probably won't change the most of the war though. The real changes are postwar. There will not be pressure for Chiang to include the Communists into a postwar government, nor will there be a rival to secure arms and territory in Manchuria. The US would still likely push for certain democratic reforms, and the Democratic League becomes the most signficant opposition to Chiang and the KMT.
The next move for Chiang would be to eliminate the remaining warlords who operate independently in Yunnan, Guangxi, Central China, and Sinkiang. He will institute payroll reforms in the army and eliminate most of the corruption there. At that point, he will begin to do some kind of economic and land reform, but run into entrenched opposition.
Without a local Communist party to prod the Soviets into giving up their gains in Manchuria, Stalin will play hardball in Manchuria. Soviet occupation will stay there longer, and Stalin will probably set up a local Communist party that will include some survivors of the original. To be perfectly honest, Soviets may never leave Manchuria, and there might be an "independent" Communist Manchurian state.
KMT China will probably run a middling course between the PRC and Taiwan. There are too many unknowns to state how far reaching and successful economic and political reforms would be, so many things are equally plausible.