Settings
In your setting, you assume you are given $n$ copies of some state $\rho$, which is either $\rho_1$ or $\rho_2$ with equal probability. The setting you have chosen is also symmetric in another way - you treat the error of misidentifying $\rho_1$ as $\rho_2$ to be the same as the error of misidentifying $\rho_2$ as $\rho_1$. You are not allowed joint measurements on $\rho^{\otimes n}$. You could consider sequential measurements where the result of the previous measurements is used for subsequent measurements but in my answer, I assume this is not allowed either. So you make $n$ independent measurements.
Many of these assumptions can be changed and some possibilities are covered in Chapter 7 of Tomamichel's book.
Experimental procedure
You perform the Helstrom measurement $n$ independent times and you obtain $s$ measurements where you get the correct result and $n-s$ measurements where you get the incorrect result.
Probability of correct guess
Your probability of correctly identifying the state is $P_w$. The probability of losing this is $P_l = 1 - P_w$. You have that
$$P_{w} = \frac{p^s(1-p)^{n-s}}{p^s(1-p)^{n-s} + (1-p)^sp^{n-s}}.$$
How many measurements?
We know that for sufficiently large $n$, $s = pn$. Substituting that and doing some algebra, you obtain
\begin{align}
\frac{1 - P_{w}}{P_{w}} &= \left(\frac{(1-p)^pp^{(1-p)}}{p^p(1-p)^{(1-p)}}\right)^n\\
\log\frac{P_{l}}{P_{w}} &= n\log\left(\frac{(1-p)^pp^{(1-p)}}{p^p(1-p)^{(1-p)}}\right) = -n\left(D(p\|q) + D(q\|p)\right)\\
P_{w} &= \frac{1}{1+2^{-n\left(D(p\|q) + D(q\|p)\right)}}
\end{align}
I have set $q = 1-p$ and used the quantity $D(p\|q)$, the Kullback-Leibler divergence.
