This is how I get a mirror shine on amber by hand:
I go up to 1000 grit sandpaper (wet), then use plain toothpaste (no colors, whiteners etc.) on smooth leather. The last step is polishing the amber DRY with the rough side of leather (I use cow). Rub it quickly and use firm pressure. Polish in sort bursts & dip in water so the amber doesn’t overheat. Of course, rinse the amber well with water between steps.
The last step was a happy experiment that worked great! It also doesn’t clog the ‘pores’ of the amber if you want to use it for healing purposes. Another tip: use a firm sponge backing when sanding and polishing. Old mouse pads work well. This helps to prevent flat spots on your piece.