All-dielectric resonant metasurfaces are expected to boost Mie resonances with high Q-factors and enhanced electromagnetic fields owing to their large mode volumes and low material losses. However, the toroidal dipole (TD) and magnetic dipole (MD) are usually suppressed by other stronger multipoles due to their relatively weak coupling to the incident lights. In this work, the double resonances of TD and MD quasi-bound state in the continuum (quasi-BIC) are excited asymmetrically by breaking the geometric symmetry in an all-dielectric metasurface consisting of arrays of silicon I-bars and silicon Φ-disks, leading to their corresponding enhanced electric field confinements and high Q-factors. The sensing performances by these two resonances are investigated as well, achieving refractive index sensitivities of 784.8 nm/RIU and 630 nm/RIU, respectively. This work suggests a route to manipulate strong TD and MD quasi-BIC excitations and facilitates their practical applications such as nonlinear light sources and sensing.