How are random numbers structure-less? I'm using random numbers for simulations. The main reason is to have an input sequence where no (simulation) algorithm is going to lock on a pattern and introduce unwanted effects into the simulation. So it seems random numbers have irreversibly lost the structure and it's unlikely that an algorithm is going to "reverse-engineer" it.

Is there a formal way to quantify this, such that pseudo-random numbers come out as being very unstructured?

I've read something about Kolmogorov complexity, but strictly speaking there is a very simple algorithm which can generate (pseudo)random numbers?! So they are not complex in this sense?
I can give several answers to your question.

Algorithmic randomness. When should we call a sequence $x_1,\ldots,x_n$ of bits random? A priori, all sequences have the same probability, so it's not clear on what grounds we should single out one sequence or another as not being random. This is partly a philosophical question, but "applied philosophy" has given rise to several definitions of algorithmic randomness, detailed in the Wikipedia entry. These notions are actually for infinite sequences, but some of them can also be used for finite sequences, for example the notion of Kolmogorov-random sequence that you mention in your question. The trouble with these notions is that one cannot compute random sequences – that's a feature of the definitions.

Pseudorandomness. Complexity theorists and (theoretical) cryptographers prefer a more permissive notion of randomness. Suppose you need a random sequence to run algorithm A which belongs to complexity class C. A pseudorandom sequence is a sequence that "fools" algorithms of complexity class C – that is, it looks random to such algorithms. More formally, algorithms of class C cannot distinguish a small set of pseudorandom sequences from a truly random sequence. As joro mentions in their comment, for this definition to make sense, you need to look at a collection of pseudorandom sequences, since one sequence could always be distinguished from a random sequence (at least in non-uniform computation models).

There are no known pseudorandom sequences against complexity classes beyond some very weak ones, though there are constructions which work assuming some complexity assumptions. Such constructions are not necessarily useful for practical applications, for three reasons: (1) you are given a large set of sequences rather than one sequence, and you have to run your (decision) algorithm on all of them; (2) constructions in complexity theory tend to be impractical; the pseudorandom sequences are probably really hard to generate in practice, especially for realistic values of $n$ (the input size); (3) asymptotic analysis doesn't usually yield concrete bounds, so it would be hard for you to figure out what parameters to use; worse, if the security of the construction relies on some non-explicit complexity assumption (e.g. P$\neq$NP), the guarantee could be purely asymptotic, and not give any explicit bounds.

Practical randomness. Random numbers are used in practice in two kinds of situations: numerical simulations and cryptography. When using random numbers for numerical simulations, the "quality" of the numbers is not as important as the time it takes to generate them; in practice, even rather simple pseudorandom number generators seem to "work". A search of the relevant literature will reveal some modern examples. Some of the old pseudorandom number generators were indeed bad (in some situations), but the this threshold has been crossed, and from an engineering perspective the problem can be considered solved.

Random numbers used for cryptographic purposes need to be more secure, since now we're against somebody who's construction the algorithm for the express purpose of reverse engineering the random stream. In this case one needs to be a bit more careful and use a secure, industry standard stream cipher. Such ciphers are available, and I strongly urge against using anything else, especially anything which you designed yourself. For extra security, XOR the output of several stream ciphers using independent keys.