How to get a big, flat surface uniform?

Mahlzeit Please excuse my wording. I’ve no experiance with technical
terms of goldsmiths, esp. not in English.

I’m currently doing silver smart cards, i.e. the body of smart cards
made of silver instead of plastic, but also with a real smart card
module embedded into it. Milling the card (0.8mm, Ag935) was no
problem thanks to the computer controlled milling cutter my father
built. (He wants to build watches.) My problem is now getting the
surface done. The current surface structure is from the rolling mill.
(Oh why is everything called “milling”?) I tried to get this streaky
surface appearance uniform (and to remove other scratches) with fine
abrasive paper and oil. The result was not good. The surface was
blotchy(?) with lighter and darker areas, probably because of
different reflection. Also regular marks from the rolling mill could
be seen. Scratching the surface with a rotating brass brush disk(?)
gave an even less uniform surface. The last try was to polish it
with polish red and a fabric disk. This was better, but then
marks/holes which are to deep to be removed by this polishing.

So what could I do to get there a regular and uniform surface?

And related questions: Is “Weisssieden” (reducing the copper content
in the surface with sulfuric acid) necessary/important with 935
silver? Is is doable without reducing the hardness of the rolling
mill process? Does it help me to get a uniform surface or does it
make it harder?

Thanks
Matthias Bruestle