We are going to use that the homotopy category $K(\mathcal{A})$ is a triangulated category, see Proposition \ref{proposition-homotopy-category-triangulated}. By Lemma \ref{lemma-subcategory-left-resolution} we can find a termwise surjective map of complexes $P_1^\bullet \to \tau_{\leq 1}K^\bullet$ which is a quasi-isomorphism such that the terms of $P_1^\bullet$ are in $\mathcal{P}$. By induction it suffices, given $P_1^\bullet, \ldots, P_n^\bullet$ to construct $P_{n + 1}^\bullet$ and the maps $P_n^\bullet \to P_{n + 1}^\bullet$ and $P_{n + 1}^\bullet \to \tau_{\leq n + 1}K^\bullet$. \medskip\noindent Choose a distinguished triangle $P_n^\bullet \to \tau_{\leq n + 1}K^\bullet \to C^\bullet \to P_n^\bullet[1]$ in $K(\mathcal{A})$. Applying Lemma \ref{lemma-subcategory-left-resolution} we choose a map of complexes $Q^\bullet \to C^\bullet$ which is a quasi-isomorphism such that the terms of $Q^\bullet$ are in $\mathcal{P}$. By the axioms of triangulated categories we may fit the composition $Q^\bullet \to C^\bullet \to P_n^\bullet[1]$ into a distinguished triangle $P_n^\bullet \to P_{n + 1}^\bullet \to Q^\bullet \to P_n^\bullet[1]$ in $K(\mathcal{A})$. By Lemma \ref{lemma-improve-distinguished-triangle-homotopy} we may and do assume $0 \to P_n^\bullet \to P_{n + 1}^\bullet \to Q^\bullet \to 0$ is a termwise split short exact sequence. This implies that the terms of $P_{n + 1}^\bullet$ are in $\mathcal{P}$ and that $P_n^\bullet \to P_{n + 1}^\bullet$ is a termwise split injection whose cokernels are in $\mathcal{P}$. By the axioms of triangulated categories we obtain a map of distinguished triangles $$ \xymatrix{ P_n^\bullet \ar[r] \ar[d] & P_{n + 1}^\bullet \ar[r] \ar[d] & Q^\bullet \ar[r] \ar[d] & P_n^\bullet[1] \ar[d] \\ P_n^\bullet \ar[r] & \tau_{\leq n + 1}K^\bullet \ar[r] & C^\bullet \ar[r] & P_n^\bullet[1] } $$ in the triangulated category $K(\mathcal{A})$. Choose an actual morphism of complexes $f : P_{n + 1}^\bullet \to \tau_{\leq n + 1}K^\bullet$. The left square of the diagram above commutes up to homotopy, but as $P_n^\bullet \to P_{n + 1}^\bullet$ is a termwise split injection we can lift the homotopy and modify our choice of $f$ to make it commute. Finally, $f$ is a quasi-isomorphism, because both $P_n^\bullet \to P_n^\bullet$ and $Q^\bullet \to C^\bullet$ are. \medskip\noindent At this point we have all the properties we want, except we don't know that the map $f : P_{n + 1}^\bullet \to \tau_{\leq n + 1}K^\bullet$ is termwise surjective. Since we have the commutative diagram $$ \xymatrix{ P_n^\bullet \ar[d] \ar[r] & P_{n + 1}^\bullet \ar[d] \\ \tau_{\leq n}K^\bullet \ar[r] & \tau_{\leq n + 1}K^\bullet } $$ of complexes, by induction hypothesis we see that $f$ is surjective on terms in all degrees except possibly $n$ and $n + 1$. Choose an object $P \in \mathcal{P}$ and a surjection $q : P \to K^n$. Consider the map $$ g : P^\bullet = (\ldots \to 0 \to P \xrightarrow{1} P \to 0 \to \ldots) \longrightarrow \tau_{\leq n + 1}K^\bullet $$ with first copy of $P$ in degree $n$ and maps given by $q$ in degree $n$ and $d_K \circ q$ in degree $n + 1$. This is a surjection in degree $n$ and the cokernel in degree $n + 1$ is $H^{n + 1}(\tau_{\leq n + 1}K^\bullet)$; to see this recall that $\tau_{\leq n + 1}K^\bullet$ has $\Ker(d_K^{n + 1})$ in degree $n + 1$. However, since $f$ is a quasi-isomorphism we know that $H^{n + 1}(f)$ is surjective. Hence after replacing  $f : P_{n + 1}^\bullet \to \tau_{\leq n + 1}K^\bullet$ by $f \oplus g : P_{n + 1}^\bullet \oplus P^\bullet \to \tau_{\leq n + 1}K^\bullet$ we win.