As a direct answer, the same mill file you use for jointing saws when sharpening them will do fine - but that's for shaping the scraper, not sharpening it.
With a card scraper, the best performance comes from having the edges of the metal sheet clean, well defined and as sharp as you can get them. That's done by establishing the shape of scraper you want by filing or grinding, then by refining the finish on the edge and the sides of the scraper right by the edge to the sort of polish you'd put on a plane iron. It's just the very corner that does the work you want so defined; the finish on the sides away from the edge is immaterial. Once you've got that, the scraper will work well enough without a burr, but turning a small burr with a burnisher will often improve cutting ability.
How to get that sharp edge? Polish both sides on a sharpening stone, and polish the edge on the same stone. Just the same principle as sharpening a plane iron. Use a medium stone to get a polish, then refine it with a fine stone. The late Charles Hayward (who knew a thing or two about woodworking) recommended doing the edge by placing the scraper between the top and base of an oilstone case, and working it up and down the side of the stone. The case kept the edge nicely at 90 degrees to the sides. Something similar can be achieved by a block of wood on a sharpening stone, with the scraper held against it and worked edge-on on the surface of the stone. Freehanding works, as long as you can keep the angle at 90 degrees.
Once you have the cutting edges dead sharp and clearly defined, you can turn a small burr if you wish, and re-turn it a few times before reshaping the edge. With a scraper so sharpened, you'll have no problem producing definite shavings. If all you get is dust, the edge is not sharply defined enough, and either edge or sides right by the edge need more work on the sharpening stones.
It's actually not as much palaver as it sounds. Once you've got a nice polish on the sides, you don't need to do much more to them. Just sharpen off the old burr and repolish the edge; the sides will only need a touch-up on the finest stone. Bit like the flat face of a plane iron - once it's in good order, most maintenance of the iron is done on the bevel, with just a quick back-off of the burr on the flat face.