So, to start at the beginning: how do we grow these things?
At the heart of every LED is a crystal. For blue LEDs, this crystal is gallium nitride (GaN). How do we grow these crystals at scale? This achievement is behind this Nobel prize, but it's far from a solved problem even today. They found out you could grow GaN by flowing hot gasses containing Ga and N on top of an artificial sapphire film, which would act as a template for the crystal to grow. The problem is that GaN crystals and sapphire crystals are slightly different sizes (the gaps between their constituent atoms is different) so they don't match up exactly, and this resulted in a lot of strain and defects in the GaN crystals–but sapphire was the best we had, and it worked. The trouble is, sapphire isn't great: it's expensive and you can only make small crystals.
The big push now is to find a way of growing GaN on top of silicon. This would make it a lot easier to grow larger crystals and would also make it a lot easier to incorporate GaN into silicon-based devices. But growing on silicon comes with its own problems, so it's even harder to grow high quality films. Specifically, cracking as the crystal cools from its growth temperature is a big problem because the thermal conductivity of silicon and GaN is so different.
There's also a push to try and grow GaN directly from a liquid, which would be more like how we grow silicon 'from scratch' (rather than growing on top of a template). This shows a lot of promise but it's a long way from commercial viability yet.
I'll try and come back and comment on some other issues later on if there's interest :)