Forget "kits". Purchase a fine artists watercolor brush. I have a collection that I've collected at art store sales. #5/0 is the most useful, but broader brushes are sometimes useful.
I use Spotone, but it is no longer made. Marshall's offers a set of dyes that are just as good.
Spotting dyes usually come in a set of six colors - you generally only need the neutral and slightly warm colors. Mix the dyes in minute quantities using an eyedropper - a bottle of dye will last a very long time (my Spotone bottles are over 30 years old). But don't get too hung up on matching colors - the major objective of spotting is to 'minimize local contrast' according to David Vestal. The objective is to make the white spots less white compared with the gray areas around them.
I mix a few drops of dye on a white plastic palette - you can buy small white plastic dishes in art supply stores that are perfect for this. I use a white plastic container that formerly held makeup. Vestal uses a china saucer that he bought at a garage sale. Let the drops dry. Then, use a brush moistened in distilled water with a drop of photoflo to pick up some of the dried dye. Smear that bit of dye on another spot in the palette to make a gray spot. Repeat this process to generate a range of gray smears from very faint to slightly grayer than black. These smears can be used many times, and can be renewed as needed.
Use magnifiers (cheap drugstore reading glasses work just fine) to examine the print carefully with strong light. Examine a spot, and note the shade of gray that surrounds it - pick up a bit of dye using your moistened brush in a shade that is slightly whiter than the surrounding gray, and with a stippling effect, apply a series of dots of dye in the spot. If your light is directional, you can use the shadow of the brush to help you get the tip of the brush exactly where you want it to be in the spot - just bring the tip of the brush together with the shadow of the tip on the spot you are trying to correct. If you use a dye smear that is slightly lighter than the surrounding gray, then you can simply progressively add more spots until the spotted dye is dark enough.
Apply dye until it becomes difficult to differentiate between the spot and the surrounding gray when viewed under bright light and with your magnifiers. Then, when you look at the print with normal light and without magnifiers, the spot will no longer be obvious. Again - the objective is to reduce local contrast so that the bright white spot doesn't stand out.
To steady your brush hand, rest it on the print, using a piece of paper towel to protect the print from oils from your hand. Have a second paper towel that you hold in your other hand and use as a blotter.
Spotting is tedious, but it causes you to get up close and personal with your print. Frankly, sometimes the challenge is to know when to stop - - - you really don't need to spot the white spaces between the silver grains making up the image.