Scratched stainless steel

First, I would not use ordinary steel wool. Small microscopic iron particles get embedded in the stainless steel surface and could lead to rust. There is stainless steel wool, but I have experience using sand paper, so I'll tell you about that.

When changing a faucet, thinking it wouldn't harm a stainless steel sink, I used a kitchen knife to scrape that crusty residue left from hard water. I had put in several noticeable shallow scrapes on the top part of my sink where everyone could see it.

I used a series of silicon carbide, wet/dry automotive sanding paper to hide my blunder. First 400 grit, then 600 grit, then 800, then 1000. The 1000 grit was actually to "polished" looking so I went back and finished with 800 grit, blending the refinished area out to the rest of the sink top surface. I sanded in the direction of the surface graining.

For your backsplash, the type of finish will determine what grit you'll finish with. If is it polished to a high lustre, then you'll have to go to finer grits. If it is polished to a mirror finish, then you'll have to eventually finish with a polishing compound.

While working on my sink, I had gravity to help me put weight behind my work (so to speak), but on your backsplash you'll have to awkwardly push sideways to work the sand paper, plus you may have cabinets in the way and a hood to clunk your head into. Depending on how much trouble it would be, you may find it easier to remove the backsplash.

I you are not familiar with finishing metal, you could practice on a piece of sheet metal first, getting used to how much pressure to apply, and sanding in straight strokes so you can match the backsplash's surface graining. If your backsplash has a mirror finish, you could try a polishing disk on a drill or a dremel tool first.

There is no cure for that!