One way to do this is with infinite loop space machines, such as the one constructed by J.P. May in "The Geometry of Iterated Loop Spaces".
The main argument is that there is a contractible operad L such that BU is an L-algebra. The operad L is the complex linear isometries operad. Then, May's method proves that BU is an infinite loop space.
The argument goes roughly like this: Let $C_n$ be the little n-cubes operad, then using May's product trick, there are maps of operads: $L \longleftarrow L\times C_n \longrightarrow C_n$, the second of which is a weak equivalence.
Now, BU is also an $L\times C_n$ algebra. May shows that if $X$ is a nice space, then the free algebra $L\times C_n (X)$ is weakly equivalent to $\Omega^n\Sigma^n X$. Now, he applies a simplicial bar construction to obtain weak equivalences:
$BU \longleftarrow B(L\times C_n, L\times C_n, BU) \longrightarrow B(\Omega^n\Sigma^n, L\times C_n, BU) \to \Omega^n B(\Sigma^n, L\times C_n, BU)$
which is the required n-fold delooping. There are many details and a consistency check left to do, but this is the basic idea.
At this point you should be able to define $K^n(X) = [X, B(\Sigma^n, L\times C_n, BU)]$.