Unfortunately the tool that should work for this, bevel, falls down in this case, because it ignores the ‘segments’ setting in vertex only mode for a 2D object. The workaround is to first solidify or extrude the 2D mesh into 3D and then use the bevel tool on the corner edge. When you have the amount of rounding you desire (use the F6 or operator panel to set the amounts), just delete the extra face.
If I have misunderstood you, and those are actually 3D objects in your screenshot, then of course the bevel tool would work just as you want.