While DNA origami is a popular and versatile platform, its structural properties are still poorly understood. In this study we use solid-state nanopores to investigate the ionic permeability and mechanical properties of DNA origami nanoplates. DNA origami nanoplates of various designs are docked onto solid-state nanopores where we subsequently measure their ionic conductance. The ionic permeability is found to be high for all origami nanoplates. We observe the conductance of docked nanoplates, relative to the bare nanopore conductance, to increase as a function of pore diameter, as well as to increase upon lowering the ionic strength. The honeycomb lattice nanoplate is found to have slightly better overall performance over other plate designs. After docking, we often observe spontaneous discrete jumps in the current, a process which can be attributed to mechanical buckling. All nanoplates show a nonlinear current-voltage dependence with a lower conductance at higher applied voltages, which we attribute to a physical bending deformation of the nanoplates under the applied force. At sufficiently high voltage (force), the nanoplates are strongly deformed and can be pulled through the nanopore. These data show that DNA origami nanoplates are typically very permeable to ions and exhibit a number of unexpected mechanical properties, which are interesting in their own right, but also need to be considered in the future design of DNA origami nanostructures.