I use Brasso. Any metal polish will work.
On the broken veneer pieces - Joe is right. New veneer is about 1/2 the depth of the original. Actually repairing the veneer is difficult. I've done it, but I wouldn't do it again.
If I was doing it again, I would buy wood filler from Rocklers (online and stores in large cities). (Their filler comes in wood colors - the photo looks like it's golden oak, but I have not seen it in person.) If the surrounding spots are loose, first, I would glue them down, so they don't break further. Get a syringe and put a tiny bit of Elmer's wood glue under the edge and then cover the repair with wax paper and place a brick on top and leave it a day or so. Then, I would try to fill the broken area as best I could with the filler, let that set and then sand and finish whatever way you decide.
Disclosure: I have not tried this. I used their wood filler on a floor and it wasn't the color I expected (red oak). I had to stain the places where I used it with a tiny paintbrush.
bkay