Just one example of the problem referenced here is the need to have all the amino acids in any given protein be either all L-chiral or R-chiral, with the overwhelming majority of all proteins being L-chiral. Although you can tilt the balance slightly using cosmic radiation, this mechanism comes nowhere near being an efficient enough filter to accomplish the job that needs to be done.
I am not a biochemist. The information in this post is accurate to the best of my knowledge, but my knowledge is admittedly quite limited. If there are any substantial errors of fact, please mention them in the comments.
Amino acids used in protein synthesis, as well as many sugars, have a "left-handed" (L-chiral) and "right-handed" (D- or R-chiral) version. Nonbiological synthesis of these chemicals tends to produce roughly equal amounts of both versions, but we have observed that most (but not all) terrestrial life typically employs left-handed amino acids which operate on right-handed sugars; all terrestrial life (with only a few exceptions) exhibits homochirality.
Even without knowing much biochemistry, it comes down to an either-or: Either there is some physical reason for the observation of homochirality, or there is no physical reason.
If there is some physical reason, then that physical reason is itself sufficient explanation for our observations. It might be the case that left-handed amino acids are better for some chemical reason, or there might have been an slight imbalance in the availability of amino acids in the pre-biotic environment. The overwhelming majority of both amino acids and sugars in the modern environment are biologically synthesized; once the pre-biotic supply was used up, the only further source was whatever sort of replicators that had already arisen.
But it might be the case that there is no physical reason, or at least no natural physical reason. In which case, we have to evaluate pure chance against some sort of "supernatural" reason. There are only 22 amino acids used, and as far as I know at least one is non-handed. Thus the probability that life would have selected all left-handed amino acids is at worst 221 : 1, or about 2 x10-6. Chance events on the order of 10-6 are kind of surprising, but nowhere near the (bullshit) 10-150 threshold proposed by Dembski.
(This is the worst case scenario; any partial physical explanation would always reduce the odds. For instance, if the earliest replicators used fewer amino acids the odds would be reduced accordingly. And if there's a physical reason to at least employ amino acids of all the same handedness, the odds go down to 50/50, left or right.)
Most importantly, even assuming arguendo the absence of any natural physical reason underlying homochirality, the hypothesis of any paranormal or supernatural "intelligent design" fails to explain homochirality. If there is no natural physical reason for homochirality to evolve, then there is no discernible reason for an "intelligent" designer to prefer homochirality: The preference is arbitrary, and the designer could have equally well have chosen any of the viable alternatives. In this case, the probability that the designer would have chosen any specific chirality is exactly the same as the probability of the observed chirality occurring by chance.
This sort of analysis defeats the argument from chance for any paranormal or supernatural hypothesis of intelligent design. For anything we observe, however unlikely, there is under naturalism either a physical reason for that observation or the observation is due to chance. If the observation is due to chance, then the absence of a physical reason entails that the supposed "preference" of the designer is arbitrary, and the probability of the specific preference is equally unlikely. Positing a designer's "hidden" knowledge does not help us: An explanation in hiding is no explanation at all.
There is, of course, a general argument that could, at least in theory, establish something of an "intelligent designer": the argument from teleology. If it could be shown that sufficiently many events went one way or another with no immediate reason to explain the particular way (or, even better, good reason to suppose the opposite way), but had a future utility that could be independently established, then we could statistically infer the presence of teleology. Note that one or two events with fortuitous future consequences would be insufficient: it's certainly possible by to sometimes make the "right" decision by chance.
And it is actually the case that the events we can reliably evaluate do not show this statistical pattern: changes do not in fact get "lucky" per future utility more often than we would suppose by chance. For this reason, "intelligent design" advocates have to dig deep into the esoterica of biochemistry to find the rare surprising event. But of course it is the very rarity of these surprising events that argues against intelligent design.