The Direction of Arrival (DOA) estimation algorithm based on sparse recovery maintains high accuracy even with low signal-to-noise ratios and limited snapshots. However, many of these algorithms depend on grid partitioning and refining the grid in two-dimensional DOA estimation significantly increases computational complexity. The Decoupled Atomic Norm Minimization(DANM) algorithm provides insights for two-dimensional gridless DOA estimation, sidestepping the computational complexity associated with grid partitioning. However, this algorithm is only suitable for uniform rectangular arrays and requires angle separation conditions. To address these limitations, especially for uniform rectangular arrays with partially damaged elements, we propose a rapid gridless two-dimensional DOA estimation algorithm of linear frequency modulation signals. The algorithm leverages the fractional Fourier transform to acquire signals with time-invariant steering vectors. By employing the Jacobi-Anger expansion formula, the received data is expanded in the fractional domain, enabling the formulation of the DANM algorithm. Additionally, to mitigate the impact of angle separation conditions on DOA parameter estimation, we introduce a sparse constraint that more closely aligns with the atomic \(\ell _0\) norm, formulating a semi-definite programming problem. This problem is iteratively addressed using the majorize-minimization method and alternating multiplier method, enhancing the precision of DOA estimation while concurrently reducing computational complexity. Simulation experiments validate the efficacy of the proposed algorithm.