We present a versatile framework to study strong existence and uniqueness for stochastic differential equations (SDEs) in Hilbert spaces with irregular drift. We consider an SDE in a separable Hilbert space $H$ \begin{equation*} dX_t= (A X_t + b(X_t))dt +(-A)^{-γ/2}dW_t,\quad X_0=x_0 \in H, \end{equation*} where $A$ is a self-adjoint negative definite operator with purely atomic spectrum, $W$ is a cylindrical Wiener process, $b$ is $α$-Hölder continuous function $H\to H$, and a nonnegative parameter $γ$ such that the stochastic convolution takes values in $H$. We show that this equation has a unique strong solution provided that $α> α^*(γ)$, with an explicit function $α^*$ that takes values in $(0,1)$ for all $γ\in[0,3)$. This substantially extends the seminal work of Da Prato and Flandoli (2010) as no structural assumption on $b$ is imposed. The range of admissible $α$ is also extended. To obtain this result, we do not use infinite-dimensional Kolmogorov equations but instead develop a new technique combining Lê's theory of stochastic sewing in Hilbert spaces, Gaussian analysis, and a method of Lasry and Lions for approximation in Hilbert spaces.