The most common method is to polish the nickel with a soft cloth. This step should be the first one, even if you think that it might do much good. The reason is that you never know, how much a soaked cloth can buff away grease spots and dirt patches. The correct method for cleaning involves a circular motion rather than hasty back and forth rubbing.
In case of a green tarnish, which is a common phenomenon in nickel plated metallic objects, WD 40 or a specialized metal cleaner can be used. The trick is to apply the solution on the stained spot, let it rest for about a minute, and then run it with a steel wool. The rubbing should be done in a circular motion. The use of steel wool can cause some problem, and the best way is to first use it at a spot on the object which is not directly visible. If the steel wool causes damage, do not use it on the rest of the object.
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Another method to remove the tarnish and other stains is to clean with a piece of cloth which has been soaked in vinegar. Make sure that extra vinegar is wringed before the cloth is used for rubbing. The acidic nature of the vinegar will help in cleaning the spots. After doing the process, wipe the object with a wet cloth to remove any traces of vinegar.
For more stubborn stains, the nickel plated object will have to be soaked in a vinegar solution, having one part vinegar and four parts water. The process lasts for few hours to take effect, and the object should be wiped with a wet cloth afterwards.
The objects can also be cleaned by using a scouring pad dipped with pure ammonia, not its variants. The object needs to be rinsed afterwards.
For more intense cleaning, make a 50-50 solution of ammonia and water in a container and place your object into it for nearly 30 minutes. Wipe the object with a wet cloth after the stains are removed.
An oven cleaner can also be used for nickel plating as well as a non-abrasive metal polish, but these methods are bit expensive and should be the last resort.