Planer blades can be tricky unless you have a jig or something to maintain the correct angle. I made this one that will sharpen 2 blades at a time using up to 2000 grit wet/dry paper on a sheet of glass. Works pretty good, but takes time. In the first pic you're looking at the bevel side. To sharpen, you just put that side down on the paper. The 2 blades must be at exactly the same depth and angle in the jig, so making the jig requires precision. If you have a 3 blade planer you could probably make something similar to handle 3 at a time.
Here it is in a different pose.