Looks like a rock solid build.
Why not just polish the bottom or mar/ scratch the bottom smooth and flat, place the LED with a reflector shining up through that into the crystal? Should be groovy, and easier.
I did consider just using the crystal as diffuser, but it wouldn’t look as cool and I’d need to design a body that would look good and hold it too. I actually considered trying to make Gandalf’s staff from Lord of the Rings, where it has that scene in the mines of Moria where he plugs in the crystal and starts shining it around.
Also I’ve been considering halving the crystals length wise with a cutting wheel, and then boring out half and half and just clamshelling it around the light, but I don’t think that would be any easier really.
Good luck with it. Sounds like a challenge…
Real salt of the earth stuff.
I’ll try to keep my posts crystal clear.
apart that I like those crystals, think you have put yourself in a lot of trouble working with them 
I have a few lights around the house which shine into translucent objects for visual effect… I find it’s easiest to just place the objects on top of a LED puck light, and the objects which work best are either cloudy inside or have a lot of internal texture.
For example (this isn’t mine):
Another light I’m fond of shines three puck lights up at an artificial bonsai tree, and I intend to paint its flower petals with glow-in-the-dark paint as soon as I get a round tuit.
Putting the LED actually inside the crystal sounds like it would be pretty difficult.
If you made a mold of one crystal you could then cast resin crystals with the led inside. Tap plastics mentions making high quality costume jewelry this way.
Reminds me of…
Interesting and ambitious build.
Best wishes on your current boring challenge. With such a high melting point, it doesn't seem like you could even heat anything to melt a hole into the crystal. Steel and even titanium have lower melting points.
Awesome Idea, love it. Hope it turns out great for you.
‘oogle “smelt quartz” — the geologists will tell you there is no such crystal form, that it’s glass, which is not crystalline — despite sellers’ claims.
Very slow work with a diamond bit under running water will put a hole in glass, but it’ll fracture easily (and look at the broken surface, another way to tell crystal from glass is how it breaks)
I’d second the suggestion to see about casting the shape you want out of clear resin.
Scotty is on the right track…the crystal only needs polishing on the bottom. It will act like a “light pipe”.
Is this taking shape?
Not really. My second crystal broke as well. I’d gotten deeper, and was trying to bore it out a bit, but it broke too. As for my other parts that I’ve been working on, I’m still waiting on some smaller reed switches to come in the mail, and some more driver components, and my first couple battery holder bodies didn’t work to well either. This is a long weekend for me, so I’m hoping to get some more work done soon.
Edit: Also I picked up fountain pens as another hobby. Not helping my time availability. 
If you can temporarily glue either of the crystals back together you can make a tap plastic casting pretty easily and casting resin drills pretty easily also.
I’m giving a lot of thought to the idea of switching to resin. It might even have better thermal conductivity than the crystals I’m using. But I do have a few more viable crystals there, so I’ll try a couple more before giving up that ghost.
Do you have a carbide grit Dremel tip? All you really need is a hollow for the led dome.
Great job, original for sure.
There is also synthetic lapidary diamonds. They can be purchased inexpensively on e-bay in various sizes. They could form a nice base of crystals at the base of your large crystals. The synthetic or man made diamond is argued to conduct heat better than natural diamond. I know they are translucent but to varying degrees so that might add depth and character. If you have access to a rock tumbler or polisher you might polish some of them as well to let more light through.
Sorry for your bad luck so far. Hang in there. I usually try to do things in a far more complex way than is useful or possible and I fear you might suffer that same curse. I keep trying to apply a bastardized version of Occam’s Razor to my attempts at building things. The most simple or least complex solution is usually much better. I have little success as this goes against my very nature. Go for plan, “B”. the idea is excellent, the implementation is the challenge.