= Solution
Let $u:X\to U_{\mathcal K}$ be continuous. The family $\{u(x)-I:x\in X\}$ is norm compact in the compact-operator ideal. For any $\varepsilon>0$ there is one finite-rank <orthogonal projection> $P$ with
$$
\sup_x\|u(x)-I-P(u(x)-I)P\|<\varepsilon.
$$
To prove it, take a finite $\varepsilon$-net of <compact operators>, approximate each by a <finite-rank operator>, and include both their ranges and adjoint ranges in $P\mathcal H$. Contractivity of compression transfers the estimate to the whole family.
Take $\varepsilon<1$ and set $g(x)=I+P(u(x)-I)P$. It is invertible because it is within distance one of the unitary $u(x)$. It has the form $g_V(x)\oplus I$ on $V\oplus V^\perp$, where $V=P\mathcal H$ is finite-dimensional and $g_V:X\to GL(V)$ is continuous. The straight segment from $u$ to $g$ stays invertible; taking its unitary polar factors supplies a <homotopy> in $U_{\mathcal K}$, since inverse square root depends continuously on the positive invertible factors and preserves identity modulo compacts.
Conversely a map $g:X\to GL(V)$ defines such a class by the continuous deformation $g(g^*g)^{-t/2}$ to its unitary polar factor, then extension by the identity on $V^\perp$. Apply the same uniform finite-rank approximation to a <homotopy> on the compact space $X\times[0,1]$. Choosing $P$ to include the original endpoint supports keeps those endpoints unchanged under compression. This proves that two finite-dimensional maps give the same class exactly when, after adding identity blocks, they are homotopic through a common finite-dimensional <general linear group>. Hence
$$
\boxed{K^1(X)=\underset{N}{\operatorname{colim}}\,[X,GL_N(\mathbb C)],}
$$
with stabilization $g\mapsto\operatorname{diag}(g,1)$.
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