You need to make a holder for the drills in some fashion. Then the trick part is to make a cam action that allows the drill bit to move in and out a little bit as it is rotated.
The Darax drill bit sharpeners do it that way to get the end geometry kind of right. They have the 2nd operation that then will cut the split point or the web thinning.
Their holder has a pair of pins, that rub against the cam profile plate.
With a loupe or looking through a microscope or similar magnification, like 10x for example, using an India stone, or a fine diamond file, you can resharpen a drill bit. I was shown that about 42 years ago. As I had the young eyes then, used a 3x loupe , the monocular type.
Even buying new drills is no guarantee that they are correctly sharpened. With the small drills, brands like TiTex are very good. There is a series that come on larger shank diameters as well. For 1mm drills they have them with 2mm shanks, and the smaller ones I have, are on a 1mm shank. Like the 0.4 drill on a 1mm shank. Sometimes you may want to drill fairly deep with a small drill and then the straight shank will be the best choice.