It is only "broken" if you assume incorrectly that the goal of science is to Discover Truth. Moreover it is broken in a more direct way: there exist certain models which are
mathematically equivalent but which describe
contradictory states of being. A reasonably good example of this is heliocentrism vs. geocentrism: classical mechanics allows you to say "the Earth is at the center of the Solar System, there are gravitational, Coriolis and centrifugal forces around it affecting all of the stuff in space", but it also allows you to say "The Sun-Jupiter barycentre is at the center of the Solar System, and the only force we need is gravity." There is no experiment which can distinguish between those two; they are mathematically equivalent.
(A slightly better example comes from quantum mechanics. In the "Schrodinger picture" there is a "wavefunction of the universe" which changes from moment to moment, while the definitions of space and momentum stay the same. In the "Heisenberg picture" the wavefunction stays the same while the definitions of space and momentum change. You would think there would be an ontological difference to the question, "is the state of the universe different from the state of the big bang?" but, in fact, on this description there is no observable difference, and science could never settle the question.)
This does not reduce science to a religion; science simply studies the observable differences and must be content with not knowing everything -- which most scientists are already content with, since they have to deal with matters of uncertainty and the distinctions between correlations and causations.
Dark Matter does interact with other things, but it does so indirectly, because it has mass and therefore warps spacetime. This is not actually the first use of gravitational lensing to observe dark matter; in fact, earlier it had been used to settle the question of whether dark matter felt any electromagnetic force at all, by looking at galaxy collisions. The prediction would be that the dark matter clouds of two galaxies would more or less "go through each other" in a collision while the luminous stuff would "bump into each other". This was observed as early as 6 years ago, see http://chandra.harvard.edu/photo/2006/1e0657/ .
We certainly can show you, and we can explain to you what it is. The only problem is the same problem that neutrinos have: it's just very hard to detect these particles because they don't have an electric charge and therefore don't care about the electrons which make all the rest of chemistry happen. Our best tool for understanding dark matter is still gravity; its a force which we know the dark matter feels.
