Stellar streams are particularly sensitive probes of the mass distribution of galaxies. In this work, we focus on the Helmi streams (HS), the remnants of an accreted dwarf galaxy orbiting the inner Milky Way. We examined their peculiar dynamical properties in depth, and used these to provide tight constraints on the Galactic potential, and specifically on its dark matter halo in the inner 20 kpc. We extracted 6D phase-space information for the HS from Gaia DR3, and confirm that the streams split up into two clumps in angular momentum space, and that these depict different degrees of phase mixing. To explain these characteristics we explored a range of Galactic potential models with a triaxial NFW halo, further constrained by rotation curve data. We find that a Galactic potential with a mildly triaxial dark matter halo, with $p=1.013^ $, $q=1.204^ $, $M_ discs M_ odot $, and $M_ DM kpc M_ odot $, is required to form two clumps in angular momentum space over time. Their formation is driven by the fact that the clumps are on different orbital families and close to an orbital resonance. This resonance also explains the different degrees of mixing observed, as well as the presence of a dynamically cold subclump (also known as S2). This first and very precise measurement of the triaxiality of the inner dark matter halo of the Galaxy uniquely reveals the high sensitivity of phase mixed streams to the exact form of the gravitational potential.