connecting an LED to a (sideglow) fiber cable
There are connectors for this, but I've never done it with such large diameter fiber. The ones I have are used to connect an LED or phototransistor to 1/16" plastic fiber.

Have you tried googling "plastic fiber connector?" Motorola (now Freescale) Optoelectronics used to carry quite a variety of them among other manufacturers. Digikey carries them. They also had/have an entire component that encased an LED in a plastic block with a compression fitting that would accept plastic fiber.

You can also make your own connector if you have access to a drill press. Drill a 3mm hole through a block of plastic/metal, flip it over and drill a hole that's a snug fit for the LED of your choice.

If you do decide to glue it, I'd suggest using a flat-top LED. However, you will probably have to get the end of the fiber perfectly flat or