LTT 1445 is a hierarchical triple M-dwarf star system located at a distance of 6.86 parsecs. The primary star LTT 1445A (0.257 M_Sun) is known to host the transiting planet LTT 1445Ab with an orbital period of 5.4 days, making it the second closest known transiting exoplanet system, and the closest one for which the host is an M dwarf. Using TESS data, we present the discovery of a second planet in the LTT 1445 system, with an orbital period of 3.1 days. We combine radial velocity measurements obtained from the five spectrographs ESPRESSO, HARPS, HIRES, MAROON-X, and PFS to establish that the new world also orbits LTT 1445A. We determine the mass and radius of LTT 1445Ab to be 2.87+/-0.25 M_Earth and 1.304^{+0.067}_{-0.060} R_Earth, consistent with an Earth-like composition. For the newly discovered LTT 1445Ac, we measure a mass of 1.54^{+0.20}_{-0.19} M_Earth and a minimum radius of 1.15 R_Earth, but we cannot determine the radius directly as the signal-to-noise of our light curve permits both grazing and non-grazing configurations. Using MEarth photometry and ground-based spectroscopy, we establish that star C (0.161 M_Sun) is likely the source of the 1.4-day rotation period, and star B (0.215 M_Sun) has a likely rotation period of 6.7 days. We estimate a probable rotation period of 85 days for LTT 1445A. Thus, this triple M-dwarf system appears to be in a special evolutionary stage where the most massive M dwarf has spun down, the intermediate mass M dwarf is in the process of spinning down, while the least massive stellar component has not yet begun to spin down.