How the heck do I get the stubborn white deposit off my bathroom ware?

This is a classic. The white stuff is likely minerals that used to be dissolved in your water. Although there are some commercial products that may be more effective, your first option is white vinegar. Pennies instead of dollars. Everyone likes to save money. Your white stuff may just wipe off with a vinegar dampened cloth or you may need to remove the shower head or faucet strainer or whatever ware and soak it in vinegar a while.

Vinegar and baking soda will really help break up that film/deposit, as well as a non-abrasive scrubbing pad for the tough spots.

CLR, apple cider vinegar or Lime Away.

Soak a dryer sheet in vinegar and lay over the top. Allowed to remain there for at least an hour before wiping away.