The Shinto deity associated with the hearth, kitchen, and controlled household fire is Kōjin.
Kōjin is also known as Sambō-Kōjin and, in household contexts, Kamado-gami—the god of the cooking stove. Unlike Kagu-tsuchi, whose fiery birth is destructive and kills his mother Izanami, Kōjin represents fire harnessed for human life. His domain includes cooking, warmth, purification, and protection of the household.
Kōjin was traditionally honored near the hearth, sometimes through a simple fuda tablet and sometimes through an elaborate temple statue. Artistic images can show him with flaming hair, fangs, multiple heads, and several arms. These fierce features reflect fire’s dangerous power as well as its protective role.
Kōjin is sometimes identified with the Buddhist deity Fudō Myō-ō, another flame-surrounded figure associated with overcoming impurity. In some traditions Kōjin also watches household behavior and reports wrongdoing to higher kami. Inari is chiefly connected with rice, prosperity, and fox messengers, while Kagutsuchi is the primordial fire deity, so neither is the best answer here.