= Solution
Extend the <slash operator for modular forms> to positive-determinant matrices by
$$
(f|_k\alpha)(\tau)
=\det(\alpha)^{k/2}(c\tau+d)^{-k}f(\alpha\tau).
$$
The <double coset>
$$
\Gamma(1)\begin{pmatrix}p&0\\0&1\end{pmatrix}\Gamma(1)
$$
has left-coset representatives
$$
\begin{pmatrix}p&0\\0&1\end{pmatrix},
\qquad
\begin{pmatrix}1&b\\0&p\end{pmatrix}
\quad(0\leq b<p).
$$
Therefore the sum of the corresponding slashes, multiplied by $p^{k/2-1}$, is exactly
$$
T_p(f)(\tau)
=p^{k-1}f(p\tau)
+\frac1p\sum_{b=0}^{p-1}f\left(\frac{\tau+b}{p}\right).
$$
Right multiplication by an element of $\Gamma(1)$ permutes these left cosets. The cocycle law for the slash operator consequently gives
$$
T_p(f)|_k\gamma=T_p(f)
\qquad(\gamma\in\Gamma(1)),
$$
so $T_p(f)$ is a weight-$k$ level-one modular function. Each displayed summand is holomorphic on $\mathfrak h$, hence so is their finite sum. This is the <Hecke operator on modular forms>.
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