One of my favorite topics. The book 'Military History of Soviergn Hawaii' by Niel Bernard Dukas I think has the best analysis and break down on this topic. He argues that while Kamehameha did have several muskets and a few cannon they did not themselves change the tide of any one battle or cause a significant loss of life that they were necessary. It was Kamehameha's organised units and formations that truly shattered all who opposed him.

But in "The Warrior King" by Richard Tregaskis he writes that: During the invasion of Oahu Kamehameha's favorite and best cannon "Lopaka" (Robert) was pushed up the muddy slopes of the Pali by a 100 man detail carrying the shot and powder and driving the Hawaiian made sled and wheels though the vegetation. The canon did not reach the front in time for the first two skirmishes but when Kalanakapuli's forces fell back to a rock wall near modern day jackass ginger pool, they formed in a tradition Hawaiian battle line of a crescent moon with the highest ranking ali'i in the center. When the canon finally made it to the front John Young ordered one solid shot fired into the center of the formation into the rock wall they were crouched behind. It is said that the first shot from the cannon killed the popular turncoat general Ka'iana which started the final route of Oahuan forces.

So overall the guns and canons were more of a moral boost than battle winning devices. But the death of Ka'iana definatly shortened the invasion and opposition on Oahu. Kamehameha's key to victory was ordering his troops to fight in pike squares and units of melee and ranged compared to the traditional individual mob combat of Hawaii/Polynesia.