The question actually boils down to the question of whether or not there is a single underlying reality. Science assumes that there is, and recognizes that what we observe is merely a shadow of a slice of that underlying reality, since we can only hope to observe a very small part of the whole world, and our observations will not be a perfect representation of the real thing. The laws of nature then refer to the rules governing the underlying reality, not what we observe or the scientific hypotheses that we make.
The problem with discarding the assumption that there is an underlying reality is that there is then no consistent worldview. You would have to grant that it is absolutely true that something exists, even if there is nothing you can say about what kind of thing exists. Since the negation (that nothing exists at all) is absolutely false, you can see that the fact that something exists is indeed one aspect of the underlying reality.
Since it is rather evident from Science (even if not absolutely provable) that the underlying reality is governed by laws, your question of what enforces the rules themselves is a very astute one. Clearly something must, otherwise there is nothing to bring about the governance in the first place. We must conclude then that the totality of all the laws governing reality must both govern and enforce itself, since anything that governs that totality must be included in it by definition. Note that this totality includes even laws that govern other laws.
From this point, the atheist concludes that the world explains or creates itself. In contrast, the theist concludes that there is a conscious lawmaker called God that made and enforces all the laws for certain purposes, who is self-explanatory (such as being called "I will be what I will be" in Abrahamic religions). The only difference is essentially whether the self-explanatory core is conscious (whatever that means). And the agnostic thinks it cannot be known. Note that all have to accept that it is perfectly consistent to have absolutely nothing at all exist, but it is simply a fact that our world does exist.
Whichever viewpoint you choose, it would be difficult to imagine just how the laws are enforced, but certainly it is not by computation in the sense that we are familiar with, and so your reason of rejecting the existence of underlying laws is flawed because you assume that something must compute that way. To give an (imperfect) analogy, a machine works in a specific way because its parts and programming forces it to work that way. If you did not see those constituents, it does not change the fact that it is governed by them, and in fact you can often infer not only the existence but the nature of those constituents to a certain extent just by observing the way the machine works. This is science. The difference is that there is some fundamental part of reality which governs itself, which science may never reach, and that is part of the next point.
The observations that we can make in our world may not be able to probe much of reality. This is almost certainly true if we use only scientific experiments, because experiments are incapable of distinguishing between many different hypotheses that are consistent with all the observations but contradict each other. This problem cannot be eliminated because there are infinitely many hypotheses that can 'explain' everything we have observed but differ in places we have not or cannot observe. For example, you could have been constructed by some laws just one second ago and implanted with memories that make it seem as if you have existed for much longer. It is consistent, but no experiment can refute it. This is where we need logic and philosophy to guide our hypotheses and so we choose the simplest ones that explain all our current knowledge.
It is highly likely (and history has repeatedly confirmed) that the underlying reality will always be more complicated than the simplest model we construct for our current knowledge, but that does not mean that we give up and do not construct any model at all just because we will never get it. We make models of the world with the main purpose of being able to predict roughly how the world will behave in the future, so that we can take appropriate actions now. For instance, we give food and water to people who depend on us because we predict that they require food and water to live. And so far, at least this necessity of nourishment seems to coincide with the underlying reality!
