First, there’s a good video to watch on Youtube to get the basic idea what you might expect inside your Kindle. I encountered this video some time ago and now it’s been more than helpful.
So to get inside you need to release a bunch of plastic clips (the picture below shows their layout). Then use a tooth brush to clear sticky glue that causes all the troubles. I've also put a tiny amout of oil in to be sure the spring mechanism will operate smoothly.

The switch now works as expected, no more troubles!