This is what I will be using the sharpen this card scraper
 - A block of wood with a thin slot cut through it that is longer than the scraper
 - A single cut file
 - A carbide burnisher
 - A fine sharpening stone (DMT Extra Fine in this case)
 - Honing oil or light machine oil

Apologies for the poor lighting. This is my "decommissioned" bench in the corner of the workshop I resign to messy tasks I don't want to contaminate my work -  like cutting metal. The lighting works well for tasks, not so much for pictures 

I start by slipping the scraper into the kerf in the block of wood. I've found the vise jaws either can't get a good enough grip on something this thin, or cause it to bow a bit. Another advantage here is I only have to loosen the vise a tiny bit to slide the scraper out to do the other side 

Now I file the old burr off. I like to do this step every time. While I could simply burnish the surface again to roll the burr, I like to have fresh metal to do so. I am using a single cut file here. Notice the notches in the files teeth are inline with the scraper. The teeth run in the opposite direction on the bottom. This will ensure I get a clean, even, cut. I hold the file flat across the surface and make a few passes

To make things easier on the next step and to make sure the scraper is flat, I run the file straight across 2 or 3 times - similar to jointing saw teeth before sharpening. Once done I flip the card scraper over and do the same to the other side 

I then smooth out the face on my XF diamond stone. I work both edges on both sides (4 faces) until the rough burr from the file is gone

Now to polish the face. I have a jointed block of wood I put on the stone to guide the scraper. I rub gently back and forth moving to different parts of the stone. Once all the file marks are gone and the surface is smooth, I flip and do the other side. It's important the face is smooth, straight and square. In fact polishing to a higher grit will result in smoother surfaces during use. 

Here is where everything used to go wrong - turning the hook with a burnisher. This method makes it foolproof.

1. First add a little oil. This will let me make gentle passes without sticking or binding anywhere.   
2. You'll see I'm choked up on the end of the burnisher. My fingers act as guides. I first place one hand (my right hand in this pic) up against the edge of the scraper, holding the burnisher at an arbitrary but about 5 degree angle.
3.  I now grab the other end of the burnisher with my left hand; keeping it up against the scraper. It acts as a guide.
4. Now I take 2 VERY light passes and check for a burr. I know I am done when my fingernail just barely catches on the edge. 
   
   Flip and repeat x3

Done. Now I get nice shavings from the scraper instead of dust. If the scraper is leaving a rough surface, try polishing the edge with a finer stone, or leaving an even smaller burr. This process takes about 3 minutes once you get the hang of it