What are the tradeoffs for the various options coupling a LED to an optical fiber?
"Typical" fiber for communication is 9, 50, or 62.5 um diameter in the core, but there is a 125 um cladding that is also necessary for the fiber to work. There is also 900 um core plastic optical fiber. What is out there for illumination, I'm not sure. Clearly you're not working with one of the types that I'm familiar with.

But, the key point is that the core is smaller than the cladding. And the light coupled into the fiber will be (best-case) what falls on the core. With .87 NA you'll probably get pretty close to this ideal. So I'll just make up some numbers. Say you have a 30 um core in a 60 um cladding. Overall the maximum coupling efficiency you could expect from uniformly illuminating the end of the fiber bundle is 302/602 or 25%. 

Then there's a factor for the packing density of round objects into a an area (the gaps between the fibers in the bundle) which I believe is about 78% best-case.

And a reflection loss of about 4% for light entering the glass from air.

Add all these up, (.25) x (.78) x (.96), and you have about 19% best-case coupling efficiency.  (You'll need to re-calculate this knowing the correct factor for the first term)

If you are getting near this, I'd say you are actually doing pretty well.

Of course it wouldn't hurt to find an LED that emits in a narrowish cone instead of over half of all space, or even to add some kind of lens to focus the light on the area of the fiber bundle. But generally your best case coupling is still not going to be all that great.