In the movie they prove that the swords aren't really necessary with the rocket punches and all.
Their resilience and strength is superior to a lot of kajjus. Instead of using a sword to cut couldn't the Jaeger just rip the body part off? Would any of the weapons like the laser cannon or missiles be more effective than the sword? I understand that most things in this movie happen for the sake of the moment where the mountain dew drinking sw@g master says "whoa that's so rad" but there are other ways to pierce or slice than with a sword, like with I don't know plasma? or Lazers? you could literally just make a constant beam of that gun and slice the kajju in half!