How do I put a soft glowing LED light inside this glossy-translucent object?

You cant do what you are trying to do with a simple shader on a single object. You need to combine shaders and geometry together.

Think about how the button you have referenced is constructed. It looks like a clear acrylic shell, with a diffuse interior surface covering the LED.

You need to try and recreate this in a physically accurate manner.

This is an exploded view showing how you might go about reconstructing a button like the one you have shown. I have a hollow glass shell (dark blue), with a solid translucent shape inside acting as a diffuser for a point light. The material shown is for the glass shell. The translucent cube is just a straight translucent shader with a slight blue colour.