Hubbard page 131 gives
Corollary 4.4.3 Denote by $\mathcal F_K(\mathbb D)$ the set of $K$-quasiconformal homeomorphisms $f : \mathbb D \rightarrow \mathbb D$ with $f(0)=0$. Then $\mathcal F_K(\mathbb D)$ is compact for the topology of uniform convergence on compact subsets.
The proof proceeds as follows:
[Proof part 1] By equicontinuity of $\mathcal F_K(\mathbb D)$ and Arzela-Ascoli, establish that the closure of $\mathcal F_K(\mathbb D)$ in the set of continuous mappings is compact in the uniform topology. Further establish all limits are homeomorphisms.
It remains to establish that the limit functions are $K$-quasiconformal. This is where I am stuck. A priori there is no reason to expect the limit functions to have derivatives in $L^2_\text{loc}$. And even if the limit functions have derivatives in $L^2_\text{loc}$, why should we expect $$ \left |\frac{\partial f_\infty}{\partial \bar z}\right | \leq k\left |\frac{\partial f_\infty}{\partial z}\right |?$$
Hubbard says:
all limits are in $C \mathcal H^1(\mathbb D)$ , since $$ \int_{\mathbb D} \left\lVert{[Df]}\right\rVert^2\leq K \int_{\mathbb D} \text{Jac} f=K\pi$$ so that the distributional derivatives of elements $f \in \mathcal F_K(\mathbb D)$ all lie in a fixed ball of $L^2(\mathbb D)$. Thus their limits (as distributions) do also.
I seem to be confused on if/how Hubbard is relating the values of $\left\lVert{[Df_n]}\right\rVert$ to the value of $\left\lVert{[Df_{\infty}]}\right\rVert$ when all we know is uniform convergence in compact subsets.
Also, I presume the stronger conclusion of $\mathcal F_K(\mathbb D)$ is compact in $C\mathcal H^1(\mathbb D)$ is false. That means there might be a sequence $f_n$ such that $f_{n_k}$ converges on compact subsets to a q.c. $f$ but no subsequence of $f_n$ converges to $f$ in $C\mathcal H^1(\mathbb D)$.