Maybe the differencing approach should work for you. Basically the idea is to reduce this series to a finite expression, using the lag operator. Let me explain with an example from Costa (2016, p.81) (If you're not familiar with the lag operator I suggest you to read the first section of the page linked above.):
We have the optimal price level equation (equivalent to your NKPC but less developed):
$$\tilde{P_{t}}=\theta\tilde{P_{t-1}}+(1-\theta)(1-\beta\theta)E_t\sum_{i=0}^{\infty}(\beta\theta)^i\tilde{MC}_{t+i}$$
Now, we can see that the difference between one period of time might be represented by:
$$(\beta\theta)^k\tilde{MC}_{t+k}-(\beta\theta)^{k+1}\tilde{MC}_{t+k+1}=(\beta\theta)^k(\tilde{MC}_{t+k}-\beta\theta\tilde{MC}_{t+k+1})$$
In terms of lag operator we could factor $\tilde{MC_t}$:
$$(\beta\theta)^k\tilde{MC}_{t+k}-(\beta\theta)^{k+1}\tilde{MC}_{t+k+1}=(\beta\theta)^k\tilde{MC_t}(1-\beta\theta L^{-1})$$
What this tells us is that we can use the same series lagged by $-1$ (same as forward by 1) to eliminate future periods from the original one. Let $S_t=E_t\sum_{i=0}^{\infty}(\beta\theta)^i\tilde{MC}_{t+i}$, then (omitting expectations operator):
$$S_t-S_{t+1}=
$$
\begin{bmatrix}
\tilde{MC}_{t}+(\beta\theta)\tilde{MC}_{t+1}+(\beta\theta)^2\tilde{MC}_{t+2}+(\beta\theta)^3\tilde{MC}_{t+3}+...\\
-\beta\theta\tilde{MC}_{t+1}-(\beta\theta)^2\tilde{MC}_{t+2}-(\beta\theta)^3\tilde{MC}_{t+3}-(\beta\theta)^4\tilde{MC}_{t+4}-...
\end{bmatrix}
$$=\tilde{MC}_t$$
Look that we managed to reduce the infinite sum into a single expression, now to replicate this in our prices equation, we need to get something similar to $S_t-S_{t+1}$ from $S_t$ only, taking advantage of lags because if we just sum $-S_{t+1}$ in both sides of the equation we would've done nothing. Then you can check that:
$$S_t-S_{t+1}=(1-\beta\theta L^{-1})S_t=\tilde{MC}_t$$
With that in mind, we just need to multiply both sides of our prices equation by $(1-\beta\theta L^{-1})$, and develop the corresponding algebra for arriving to:
$$\tilde{P_t}-\beta\theta E_t\tilde{P_{t+1}}=(1-\theta)(1-\beta\theta)\tilde{MC}_t-\theta^2\beta\tilde{P}_t+\theta\tilde{P}_{t-1}$$
Which allows us to make better analysis by getting rid of the series, and therefore being able to write in terms of matrices and vectors that are more "readable". I'm pretty sure you can do this same process with your equation that is in term of inflation (instead of prices). Hope this works.
