Super-resolution structured illumination microscopy (SR-SIM) offers numerous advantages such as high temporal resolution, low photobleaching and phototoxicity, and no special requirements for fluorescent probes. It is particularly suitable for long-term SR imaging of living cells. By using two-dimensional lattice structured light serving as illumination, SR-SIM can achieve faster imaging speed and reduce phototoxicity, however, it is accompanied with system complexity increasing. To address this problem, in this work, we propose a fast SR lattice structured illumination microscopy imaging method based on a digital micromirror device (DMD), called DMD-Lattice-SIM. This method utilizes a DMD and synchronous time-sharing triggering with sCMOS to generate two-dimensional orthogonal lattice structured light. The proposed method only requires the collection of five phase-shifted raw images for SR image reconstruction, reducing the acquisition time by approximately 44.4% compared with the traditional SR-SIM method that requires nine phase-shifted raw images. In this work, we also introduce a rapid SR image reconstruction method called Lattice-JSFR-SIM, which combines the advantages of joint space and frequency reconstruction (JSFR)-SIM and Lattice-SIM. The raw images are pre-filtered in the frequency domain and then undergo SR reconstruction in the spatial domain. This approach reduces reconstruction time by approximately 55.6% compared with traditional frequency domain image reconstruction processing, within an imaging field of view of 512 pixels×512 pixels. The feasibility of the proposed method is demonstrated through experiments on cell microtubules and the observation of mitochondrial division and fusion in living cells. The findings presented in this paper hold great significance and application value for enabling real-time SR imaging of living cells.