This was my first time using a card scraper and it was very troublesome to sharpen. I spent a number of hours trying different techniques, but the way that worked for me was first I lapped the front and back on my Veritas Power Sharpener just like a chisel.
I then polished the edge on a granite tile that has 5 grades of sandpaper going from 220 to 1500. To guide the edge I used a piece of angle aluminum, but anything that would keep the edge at 90 degrees would work.
Lastly, I burnish the edge by running a chrome moly shaft at a slight angle along the edge on both sides. This takes very little pressure and you can actually feel the metal edge turning over. I’ve seen the recommendation of sharpening one side to a 45 degree angle, but I could not get this to work on my thinner scrapers. I may try later with my thicker ones but for now the above technique works well enough for me. Note: the angle of the burnisher isn’t correct in the picture, the burnisher is usually more horizontal.
When good results are finally achieved with a scraper it becomes a wonderful tool to use. Much like a sharp plane its oddly satisfying to peel off nice thin little strips of wood. It is also a very good hand workout.