that purple polish is probably the go if they are in good condition to start with

if you get a good shine going i think it would be best to hit them with a clear coat paint after that to prevent needing the regular polish

i did claire's set with an air tool, using a resin type sanding disc, it was too aggressive, i then worked through 400,800,1200 wet and dry sand paper. They did look good when done although there was some odd shapes due to the air tool. But they don't look great anymore as i never touched them again after