Use 10 inch network patch panel with 19 inch switch
The most obvious thing is to get a patch panel designed for a 19" rack or one that comes with an adapter plate. 

If you have your heart set on using the patch panel that you have then you would be faced with creating something of your own. I believe that the easiest way to do that is to purchase a ready made blank 19" rack panel that is higher than your patch panel. You then cut out a rectangular hole in this blank panel to fit your patch panel which you can push through from the front and secure in place with machine screws through the existing ears.

BTW. Blank patch panels can be had in steel and aluminum materials. You may find one type to be easier to work with depending upon available tools.