What should I use to give the bathroom a really solid cleaning?

I would start with TSP- used to prep for paint. Careful NOT to mix any product containing bleach, with anything containing ammonia- it will make a deadly gas.

If it is that bad, I would use TSP, that stuff cuts through anything.

My daughter works for a hotel supply company. A representative gave them a demonstration of using Spic N'Span along with a Magic Eraser to clean. Amazing results! I used on my cast iron tub that was installed in 1950 and it was clean, clean, clean. I have tried countless other ideas and this combination was the only thing that worked.