Get a can of Goop hand cleaner and a scrub brush (the white can Goop with no pumice). It will lift all the dirt from that finish like you wouldn't believe. For the scratches and chips, get a flake of the finish off the set and match it up with some hobby paint, and carefully touch up the affected areas.
In something like this, chips draw the eye because of the difference in color. If you can come close in touching up, they virtually disappear. This is how I repaired deep scratches in my linoleum floor in my kitchen, and they're invisible unless you really look for em
