There are lots of different sources for light pipe, and light pipe from different sources acts differently. This attachment method requires light pipe that melts at a fairly low temperature. These LED shoelaces work wonderfully for this.
I also tried using good quality LED light pipe from AliExpress. This stuff worked beautifully in that it carries light really far -- this light pipe looks gorgeous for a couple meters whereas the shoelaces really only look good for about 2 feet. But.. the good stuff doesn't melt the same way.
If your light pipe doesn't want to melt, you can 3d print a case that will work just about as well:
.. or you can order one printed for you from Shapeways.
If you're going this route, check out this guide about making a 3d printed case that holds the light pipe in place. If you've got the melty kind, read on!