If its brushed nickel than it is best to try and clean it first since a new one would most likely need to be special ordered.
Removing it is very simple. Just turn the spout counter clockwise. It's hard to tell you what to dip it in because it depends on the cleaners you have available in your area and the type of spout (plastic , metal). You could start off with soaking it it vinegar to dissolve mineral deposits. It will take a while (overnight) but it should be safer to the finish. Commercial lime removers like CLR will work much faster but READ THE LABEL on how to use it properly and what finishes it's safe on.
I suppose you could also try spraying a product like "Lime Away" up the spout and letting it soak a bit if you don't want to remove it. Again, READ THE LABEL to see if it is safe for the finish of the spout.
I agree it is more of an annoyance than a serious problem. It will happen again in a few months after you clean it and will become a maintenance issue. I've learned to live with it just like everyone else.
Addendum: You didn't mention what kind of tile you have in the bath so be VERY careful if it is any type of natural stone (marble, travertine, etc.). These types of acid based chemicals will damage the finish of the tiles.