The reason your first attempt didn't work is that baking soda and vinegar together neutralize each other, and create salt water and carbon dioxide. Use vinegar by itself, and if there are a lot of these stains, don't dilute it. Let the vinegar sit in the bottle overnight. If that doesn't work well enough, try CLR. Vinegar is an acid and will dissolve hard water deposit; CLR is a stronger acid so it will do the same thing, but faster and with heavier scale.
Vinegar removes hard water spots. Let it soak for a while then rinse.
Use about a teaspoon of dishwasher soap (like Cascade) and some hot water (about half full)-- shake it up and see if that will cut it. If it helped, but didn't get it all-- do it once more. This time let it sit over night.
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