Well, it's an interesting question, but there's a bit of a problem with it. The term "kind" is a reference to the Genesis creation account, which is relevant to Creationists but not to ID Proponents. ID doesn't use the term "kind" or have any specific concept of what that might be, nor do they care to. Creationism proceeds from the Biblical account to a conclusion about reality. ID looks at the world and infers an intelligent cause for certain features of it without the Bible ever entering into it.
Now, what would Creationists (both Young-Earth and Old-Earth) consider to be a "kind"? Probably something roughly around the level of the Family classification (not Genus or Species). Of course, it must be kept in mind that the taxonomic classification system is ultimately an arbitrary construct laid on top of biological reality, so there's not much reason to think that a "kind" would in all cases have to be precisely equivalent to the Family classification. The main idea behind a creation-type model is that morphological disparity among living organisms will precede morphological diversity and that the subsequent diversity arises primarily through the *loss* of genetic information and function over successive generations.
One final comment about the original question. You said:
"For example, the biological definition of species says that two organisms are the same species if they are capable of producing fertile offspring. I can use that definition to determine whether or not two organisms I have are the same species."
This isn't entirely correct.
First of all, yes, you can use any definition to group organisms that match that definition, but the process is somewhat arbitrary. It doesn't necessarily tell us anything objectively true about biological reality other than how we've chosen to define groups of organisms.
Second, a biological species is typically defined as a reproductively isolated group, but this does not necessarily mean that the group is not capable of producing fertile offspring with another group. There is no limit placed on the *reason* for the reproductive isolation. For example, two groups of virtually identical fish swimming in the same body of water but at slightly different depths will typically preferentially mate with those who swim at the same depth. In such a case, the two groups will be classified as separate species, even though it is known that they can produce fertile offspring. Also, a single population of organisms that get split into two groups as a result of some natural disaster will become reproductively isolated from each other and thus become two biological species. It is not *necessary* for there to be any significant morphological or genetic difference between two groups, or for them to be incapable of producing fertile offspring, in order for them to be identified as different biological species. So, like I was saying, even this particular definition of 'species' that you prefer doesn't really give us an objectively accurate place to mark the distinction between types of organisms at the species level. So far, the best suggestion I've heard of for an objective dividing line / definition of species is to organize them based on their orphan genes (i.e. protein coding genes they uniquely possess, not being homologous to genes in other organisms).