It shows that only animals of the same genetic code can reproduce. Dogs cannot reproduce with cats, rabbits cannot reproduce with guinea pigs
the evidence is that "species are only related within these distinct kinds" as you said above.
As others have pointed out speciation events have been observed, and also it is well documented which specific species are capable of hybridisation and how effective i is. Just because some cat species, for example, can interbreed does not mean they all can.
However, I did not bring up this example to discuss species relationships or finally obtain an actually useful definition of 'kind', as that would be off topic. Rather I brought this up as an example of how ID/creationism limits the scientific endeavour to try and understand the world around us. So basically we can study kinds to understand the first cat kind, but no further. A limit has been placed, so that what we find is not developed from some pre-existing common ancestor of cats and dogs, but is instead the product of design decisions by a creator. So on that basis, how did those first kinds get created, what decisions went into this, and can we glean what thought processes were involved from what we see in the physical evidence? For example, why did all mammal kinds have the same backwards retina in the eye, yet three different methods of reproduction?
As far as I can see, these sorts of questions are beyond what physical evidence can show us. Of course, anyone can make conjectures about answers to questions such as these, but without anything to back them up, all such conjectures are equally valid (or invalid), so how do you suggest we proceed.
To me there seems to be an inherently defeatist attitude in the ID'ist view of science. First there is this idea of a creator which can intervene and alter the physical evidence, or completely fabricate it as in the case of 'kinds', and therefore invalidate it. There is also this suggestion that we really don't know anything, that our current theories are probably wrong, and should be discarded unless they can be shown to be 100% accurate (at least for the theories that question the genesis account), and that any changes in sciences view of the world is somehow a failure. Of course science itself operates on a degree of scepticism but we still work with all the available evidence and if a theory is invalidated that just means it will be replaced with something even more accurate. It is a refinement or improvement, not a failure.