What would have been a more efficient way to kill the Kaiju?

More efficient than a Jaeger?...Hard to say as the point of the Jaeger was to minimize the amount of toxic KAIJU BLUE made it to the shoreline and inhabited areas, so the jaegers were mostly deployed at sea before the Kaiju made landfall. They also mostly engaged in meelee combat as not to create lacerative wounds to spread Kaiju Blue-so they stuck with blunt force attacks...other than that, I guess squadrons of thousands of F-22 RAPTORS.

One missile. If an unstable two-legged robot can beat a monster with fists and such, a missile would have obliterated on of those.

The Hammer of Dawn from Gears of War should do the trick, but I'll likely need to be beefed up a bit to make sure it gets through that category 5 and up Kaiju.

If everyone remembers, that last cat 5 Kaiju survived a head on nuclear explosion.

http://gearsofwar.wikia.com/wiki/Hammer_of_Dawn

Chemical or biological warfare. Virus, cancer, drugs, poison, anything that dissolves, burns, destabilizes, fuses or destroys organic material... Sending airborne pathogens into the other world would be a sure way to destroy them all...

To see or not to see that is the question...

Of course there are other cheaper ways of fighting the Kaiju, but they're not as effective...other people have suggested lasers and nuclear bombs, but those are less effective and the capabilities are very limited. Firstly you need to be able to fight underwater and lasers are useless underwater. Lasers, bombs and missiles could work when fighting the Kaijus on land but as I said the capabilities are very limited, you could still miss the Kaijus and these weapons are not flexible...if you miss your target these bombs, missiles and lasers would just destroy the surrounding buildings and structures causing massive structural and human damage and you would end up causing more harm than good. Similar problem could happen at sea too.
Also, of you want to destroy something, destroy it at the source...just damaging the surface would not solve the problem...just fighting the Kaijus would not destroy them for good, those aliens that created the Kaijus would just continue to send more and more Kaiju one after another and eventually us humans would not be able to keep up and all our resources would dry up. Hence in the movie the humans wanted to destroy the Kaijus at the source which was the bridge between the human world and the Kaiju world, to prevent anymore Kaijus from passing through the bridge into the human world. But that would've been difficult for the humans to do with missiles and bombs...I think some ppl may have forgotten that only the Kaijus are able to pass through the bridge because of their genetic code...anything else would've just deflected off it like it always had in the past. How would the humans be able to ride a Kaiju into the bridge whilst attaching a bomb to it? It's difficult to do.

Although the Jaegers were very expensive to make but they were the most effective way to fight the Kaijus, and since they were basically giant human-like robots piloted by real humans, the Jaegers were able to fight the Kaijus with much more precision and flexibility, which lasers, bombs and other artillery lack.
Also I would think that before the Jaeger problem was even created, the humans would've tried every possible way they could think of to fight the Kaijus, this movie is set in the near future so I think they would've tried using ordinary military grade weapons and artillery. My guess is that all these weapons and methods would've been ineffective against the Kaijus, especially as they evolved, so that prompted the Jaeger program to be created...I doubt the Jaegers were the humans first plan of defense...it would make more sense if the Jaegers were created because most if not all other methods had failed.

https://www.youtube.com/watch?v=TmhuQqPcOtY

And that's only a single destroyer. 90 cell missile launch system, a 5" gun, CIWS, 25mm cannon, 30 knots+.

62 vessels in service. All of them are armed to the teeth.

We also have Cruisers https://en.wikipedia.org/wiki/Ticonderoga-class_cruiser
Too many missiles to speak of.

That's just the tip of the iceberg. Carriers stationed hundreds of miles away for safety loaded down with ordinance that's just begging to blow something up. B52's that can loiter forever just itching for a drop. B1's that can go like stink and get to the appropriate area and out in seconds. A10's that are armed to the teeth and can lurk forever.

To go over the top, you could re-commission the Iowa class battleships and fill the magazines with as much 16" as you can muster. You can fire from 20 miles out so there's no worry about them getting you. There's over-watch from Navy and Airforce so they'll let you know what's what and you can go well over 35 knots and bug out if you need to.

The point being, "Giant machines that beat up monsters, the movie" is awesome! Don't ever change Pacific Rim!