Very simple reduction of how to conquer the springs:
When at the top platform, preparing to get into position near Kong - foot, foot
When in position near Kong, preparing to run up the ladder - feet, balls
Preparing to get into position near Kong - watch for a spring to land at Kong's foot, then watch for the next spring to land at Kong's foot. If it lands at Kong's foot, go. If it doesn't, wait for the next spring
Preparing to run up the ladder - watch for a spring to land at Kong's foot, then watch for the next spring to land under his balls. If it lands under his balls, go as soon as the spring that landed at his foot is over your head.
Ultimately, it just takes a lot of practice, but if you've got the basic technique, it gets easier with every try. There's certainly a lot more in-depth information at the thread linked above, and you can see there how far you can cheat forward safely, once you are in position near Kong preparing to go up the ladder.
And don't worry, you think the springs are your last major hurdle, but once you conquer them, you will realize barrels are the real problem.