Danzō, rather than using the sharingan's ability, used the Daimyō's indecisive characteristic to his advantage - knowing he would side with whoever convinced him the most.
The Fire Daimyō has a fickle and indecisive nature and is easily swayed in his decisions through biases as well as external influences. When situations call for fast, important decisions, he makes his advisers do the whole work: the one who convinces him most is the "winner"
Danzō convinced the Daimyō that Kakashi, who was initially considered for becoming the 6th Hokage, maintained the same ideology that put Konoha in it's current state of crisis.
However, Danzō points out that the teachings that were passed down to Kakashi were the cause of Konoha's problems and current crisis. The Konoha Crush, the frequent confrontations with Akatsuki, the defection of Sasuke Uchiha, and the destruction of Konoha's infrastructure, all stemmed from the ideology that kindness and unity would lead to peace. Danzō believed the ideology simply made Konoha appear weak and left them exposed to attacks
The Daimyō believed and agreed with Danzō
The daimyō, persuaded by Danzō's words and reasoning, appointed Danzō as the Sixth Hokage Candidate, just in time for the Five Kage Summit.