The list of English conjunctions that start with K is empty. No standard conjunction, whether coordinating, subordinating, or correlative, begins with this letter. The seven coordinating conjunctions are for, and, nor, but, or, yet, and so. Common subordinating conjunctions such as although, because, if, unless, until, and while also skip the letter K. K is one of the rarer letters in English, so a gap here is not a surprise.
The main reason is how conjunctions form. Most of them are short, very old function words that Old English developed to join ideas, and none of them happened to start with the K sound. Words that do start with K, like king, kind, knife, and know, mostly name things or describe actions. Those are nouns, adjectives, and verbs, which are different parts of speech. Conjunctions also form a small, closed class. English rarely adds new ones, so a new K word is unlikely to appear.
A few phrases come close but do not count. “Kinda” and “kind of” appear in casual speech, but they work as adverbs or hedges, not as joiners of clauses. Phrases like “knowing that” and “keeping in mind that” can introduce a clause, but they are participle phrases, not true conjunctions. If you are looking for words that connect ideas, the best options are the common conjunctions that start with other letters, such as “because,” “while,” or “unless.”