an interjection of surprise
豈 is an archaic kanji used as an interjection of surprise, similar to an exclamation like "how!" or "what!" in rhetorical questions. Its standalone meaning is limited to this expressive function.
豈 originally depicted a drum on a stand, used in ancient rituals, and later came to be used phonetically for an interjection of surprise. The modern form combines 山 (mountain) and 豆 (bean), but these are not the original semantic components.
Imagine a mountain (山) suddenly sprouting beans (豆) — an unexpected sight that makes you exclaim in surprise: "豈!"
For カイ, picture a surprised shout echoing off a mountain: "KAI!" The sound カイ matches the exclamation, and the mountain 山 in the kanji reinforces the scene.