Is there a way to build a stronger LEGO ball than what was done by the Mythbusters?
There is a problem with the Mythbusters building technique.

One of the first things you learn about building with Lego is that, if you want a stable structure, your bricks need to interlock as much as possible.

The Unofficial Lego Builder's Guide illustrates this rather well:



Unfortunately, to make the build easier the Mythbustes team assembled a series of large blocks and then assembled these meta-blocks into a ball. The problem with this approach is that it creates lines of weakness between the meta-blocks.

In the shots of the ball falling apart you can clealy see these sub-assemblies inside the ball, which suggests that it did indeed break up along the lines of weakness between them.

For maximum strength, the Mythbusters team should have built the ball in layers of single bricks, ensuring that in each layers bricks bound the layer below together as much as possible.

All that said, I don't think that the faulty building technique really invaidates the Mythbusters conclusions. There is simply no way that a Lego ball of that size would remain intact. You would expect, for example, that the outermost bricks would fall away in exactly the way they did in the Mythbusters. And in fairness, building the ball with fully interlocking bricks would have taken considerably longer than doing so the way they did. 

Of course, if they're willing to supply the bricks, I'm sure we could try to do it better for them!