$\newcommand\ep\varepsilon$According to the Theorem on p. 129 of Uspensky, for all $n\ge100$ and all integers $k$ we have $$f(n,k)=2^n(\Phi(z_{n,k})+\ep_{n,k}),$$ where $\Phi$ is the standard normal cumulative distribution function, $$z_{n,k}:=\frac2{\sqrt n}\Big(k-\frac{n-1}2\Big),$$ and $$|\ep_{n,k}|<\frac{0.52}n+e^{-3\sqrt n/4}.$$
In particular, if $k\approx n/2$, then $\Phi(z_{n,k})\approx1/2$ and hence the relative error of the approximation of $f(n,k)$ by $2^n\Phi(z_{n,k})$ is no more than $\approx\frac{1.04}n+2e^{-3\sqrt n/4}$.
Approximations of $f(n,k)$ with relative errors $o(1/n^p)$ for (say) $k=n/2+O(\sqrt n)$ for arbitrary real $p>0$ follow immediately from asymptotic expansions of the probability $f(n,k)/2^n$ (that a binomial random variable with parameters $n,1/2$ takes a value $\le k$) such as Theorem 6, Ch. VI.