S.-
the mildest possible approach is to use a combination of heat and patience. use a soft bristle brush and dawn dishwashing liquid, apply to the grease spots. then get some tea towels rags or wash cloths and soak them in the hottest water you can tolerate. cover the enamel with these (wet but not dripping). Give it 15 minutes and use the rags to wipe away whatever grease has lifted. Try again.
If you don't have the heart to do it all at once, you can do this every night for several nights and eventually it will come clean assuming he isn't making a pig mess of it all the time.
As for the grids, I lay a towel down in the bath (so as not to scratch the tub) put the grids on top, smear with dawn, and soak in hot water overnight.
Best,
F. B.