I am thinking of writing a critical analysis on "The Curious Republic of Gondour" and I was wondering what literary elements are most present in the short story and would be best to cover in the critical analysis. Thanks!

The Curious Republic of Gondour is an excellent example of Mark Twain’s political satire. It is important to note that Twain’s perspectives were quite progressive for the late nineteenth and early twentieth centuries. In the decades following the Civil War, plutocracy, nepotism, and unchecked capitalism lead to political corruption in America. This story was published in 1875, before the reforms and social activism of the Progressive Era spanning from 1890-1920.
Twain's story presents a Utopian merit-based society in which women hold elevated political positions. Twain was an early supporter of the suffrage movement, discussing women’s right to vote over half a century before the Nineteenth Amendment of the US Constitution granted American women this right. In the analysis of this piece, I recommend consulting historical sources to enhance your perspective on political satire and Utopian literature in the context of Twain’s society.
You could also discuss the significance of first-person narration in Twain’s story. The narrator provides an outside perspective on Gondour’s society, describing it impartially before concluding, with a metaphor, that he prefers the political system of his own country:

The Gondour national airs were forever dinning in my ears; therefore I was glad to leave that country and come back to my dear native land, where one never hears that sort of music.

I’m not sure what the requirements are for your critical analysis; if appropriate, you could compare Twain’s political satire to that of prominent authors in his time including Ambrose Bierce and William Sydney Porter.