helmet, bowl, basin
盔 centers on a protective or containing vessel: a helmet that shields the head, or a bowl/basin that holds things. The meanings share the idea of a concave, enclosing object.
盔 combines 灰 (ashes) and 皿 (dish), but the historical origin is uncertain. It may have originally referred to a vessel for ashes, later extended to a helmet or bowl.
Picture a dish (皿) covered with ashes (灰) from a fire. You need a helmet to protect your head from falling embers, or a bowl to scoop up the ashes.
For カイ, imagine a knight wearing a helmet and holding a bowl, shouting 'Kai!' as a battle cry: Kai -> カイ.