Repair chipped marble countertop

This is a tough one. There is no easy way to remedy your problem. The best way would be to remove and replace the tiles properly. If in fact the tiles are real marble, you can grind them to create the proper angle and slowly refinish the surface by stepping down grits and finally getting to a wet polish grit of 2400. That is actually impracticable for a DIY if you don't have the equipment and materials to do it. If it were me, I'd remove a couple of tiles and reset them to fix your drainage problem. A couple of hacks with a hammer, scrape off some of the thinset, replace the tiles with a little premix thinset, regrout, wait a couple of days, seal the grout and your done. The actual labor time may be only a couple of hours, even though you have to wait a bit to use the shower again.

You would normally clean marble with etch spots with hydrogen peroxide or bleach and then repolish. From the sound of it yours may be more severely damaged. I would try both on a small spot and see how it goes.

If your marble is actually corroded then you will have to polish until you hit uncorroded marble.