Intelligent reflecting surface (IRS) is expected to play a pivotal role in future wireless sensing networks owing to its potential for high-resolution and high-accuracy sensing. In this work, we investigate a multi-target direction-of-arrival (DoA) estimation problem in a semi-passive IRS-assisted sensing system, where IRS reflecting elements (REs) reflect signals from the base station to targets, and IRS sensing elements (SEs) estimate DoA based on echo signals reflected by the targets. {First of all, instead of solely relying on IRS SEs for DoA estimation as done in the existing literature, this work fully exploits the DoA information embedded in both IRS REs and SEs matrices via the atomic norm minimization (ANM) scheme. Subsequently, the Cramér-Rao bound for DoA estimation is derived, revealing an inverse proportionality to $MN^3+NM^3$ under the case of identity covariance matrix of the IRS measurement matrix and a single target, where $M$ and $N$ are the number of IRS SEs and REs, respectively. Finally, extensive numerical results substantiate the superior accuracy and resolution performance of the proposed ANM-based DoA estimation method over representative baselines.