I think the emission plane would need to be outside the button object to work. The SSS shader is a surface shader and may not know about light being emitted from inside the object. Try moving the emission plane to outside the button object to see if that helps.
The other option might be to use a full volumetric SSS shader like the one I created here. you can get it to look very similar to the SSS shader, with the advantage that it’s fully volumetric and as such should be able to handle emission from within (it is likely to be a bit slower to render however - but that may not be an issue if the button is relatively small in your scene)