= Solution
Suppose inductively that
$$
x_{t-1}\mid W_{t-1}\sim
N(\widehat x_{t-1},V_{t-1}).
$$
The observation innovation is
$$
y_t-\widehat x_{t-1}
=(x_{t-1}-\widehat x_{t-1})+\eta_{t-1},
$$
so its conditional variance is $V_{t-1}+1$ and its conditional covariance with $x_{t-1}$ is $V_{t-1}$. The <scalar static-state Kalman update> therefore gives
$$
\boxed{h_t=\frac{V_{t-1}}{V_{t-1}+1}},
\qquad
V_t=\frac{V_{t-1}}{V_{t-1}+1}.
$$
Adding the known control $u_{t-1}$ shifts the conditional mean but not the variance, giving exactly
$$
\widehat x_t
=\widehat x_{t-1}+u_{t-1}
+h_t(y_t-\widehat x_{t-1}).
$$
Since $V_0=1$, induction yields
$$
\boxed{V_t=\frac1{t+1},\qquad h_t=\frac1{t+1}.}
$$
Every update is an affine conditioning operation on jointly <Gaussian random variables>, so
$$
\boxed{x_t\mid W_t\sim N(\widehat x_t,1/(t+1)).}
$$
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