= Solution
Put $T=t_{\mathrm{mix}}^{(2)}(a,1/4)$ and $m=\lceil t_{\mathrm{rel}}\rceil$. The expected local time is
$$
\mathbb E_a\sum_{k=0}^{T-1}\mathbf1_{\{X_k=a\}}
=T\pi(a)+\sum_{k=0}^{T-1}\{P^k(a,a)-\pi(a)\}.
$$
Part c and the supplied return identity give
$$
T\pi(a)\leq8\pi(a)\mathbb E_\pi\tau_a
=8\sum_{k=0}^\infty\{P^k(a,a)-\pi(a)\}.
$$
The second term is bounded by the same infinite sum. Part b now yields
$$
\mathbb E_a\sum_{k=0}^{T-1}\mathbf1_{\{X_k=a\}}
\leq\frac{9e}{e-1}
\sum_{k=0}^m\{P^k(a,a)-\pi(a)\}
\leq\frac{9e}{e-1}\mathbb E_a\sum_{k=0}^m\mathbf1_{\{X_k=a\}}.
$$
Thus the hinted universal constant $C=9e/(e-1)$ works.
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