Consider a fixed-for-floating swap with reset dates $T_0, \ldots, T_{n-1}$ and payment dates
$T_1, \ldots, T_n$, where $0<T_0 < \cdots < T_n$. We assume that the swap exchanges the floating rate payments $L(T_{i-1}; T_{i-1}, T_i)\Delta T_i$ and the fixed rate payments $K\Delta T_i$, for $i=1, \ldots, n$, where $\Delta T_i = T_i -T_{i-1}$.
The value of the swap at time $t$, where $0 \leq t \leq T_0$, is given by
\begin{align*}
\sum_{i=1}^n L(t; T_{i-1}, T_i)\times \Delta T_i\times P(t, T_i) - K \sum_{i=1}^n P(t, T_i)\times \Delta T_i,\tag{1}
\end{align*}
where $P(t, u)$ is the price (i.e., zero price) of a zero-coupon bond with maturity $u$ and unit notional.
The forward swap rate $s$ is the rate $K$ such that the swap value given by $(1)$ is zero, that is,
\begin{align*}
s = \frac{\sum_{i=1}^n P(t, T_i) \times L(t; T_{i-1}, T_i) \Delta T_i }{\sum_{i=1}^n P(t, T_i) \times \Delta T_i}.
\end{align*}
For a quarterly resetting frequency, that is, $\Delta T_i=\frac{1}{4}$, then $\frac{1}{4}L(T_{i-1}, T_i)$ is the quarterly interest rate. Moreover, the quarterly swap rate is given by
\begin{align*}
\frac{s}{4} = \frac{\sum_{i=1}^n P(t, T_i) \times \frac{1}{4}L(t; T_{i-1}, T_i)}{\sum_{i=1}^n P(t, T_i)}.\tag{2}
\end{align*}
For the given example, we have that $n=2$, $P(t, T_1) = 0.998566$, $P(t, T_2) = 0.99655$, $\frac{1}{4}L(t; T_0, T_1) = 0.0014358$, and $\frac{1}{4}L(t;T_1, T_2) = 0.0020222$. Based on Formula $(2)$, the quarterly swap rate is given by
\begin{align*}
\frac{s}{4} = \frac{0.998566 \times 0.0014358 + 0.99655 \times 0.0020222}{0.998566+0.99655} = 0.17287\,\%.
\end{align*}
The computation for swap rate with semi-annual or annual accrual frequency can be proceeded analogously. Moreover, you can set $T_0$ to any date you like, for example, September or December, as you mentioned.
