How can I extract menthol powder from fresh mint/peppermint?

This is a fun question, but your plan is not very practical.

If you indeed insist on doing it, you have three steps in the process of menthol extraction:

1. Extract the essential oil from the leaves
2. Separate the essential oil from the solvent you used
3. Freeze the menthol out of the essential oil.

For the essential oil extraction, the only method doable at home is a steam/water distillation. (OK, there are also absolue methods, but they are less practical, and not necessary for mint). So you are left with a mixture of a hydrosol and mint essential oil.

Here is where it becomes difficult: the second step is to separate the oil from the hydrosol. Most methods I know of will produce something which is not food-safe. The popular ones are to use a nonpolar solvent and then boil it off (I hope you don't plan on handling diethyl ether in your kitchen!) but I also found a reference to "passing it through a bed of anhydrous sodium sulphate". This is probably your best bet edibilitywise, but getting the supplies and learning the process won't be easy.

In the end, you have to freeze the essential oil to a sufficiently low temperature (the source with the sodium sulphate uses -60 C but I also found references to -22), then keep it for a few days and harvest the crystals.

If you are brave enough to do it at home (and live in a place where it is not illegal to have a distillation still, and found a freezer that goes to -22) you will need time, space for the apparatus, and mint. Lots of mint. Here I found an article for a portable still which weighs 635 kg, has a 500 l distillation tank, can process up to 9 kg dried mint per distillation (you'll first need an area where you can hang 72 kg of fresh mint to dry out for a month or two, or find a dehydrator and pay for the energy to process it) and yields about 100 ml of essential oil from peppermint, half that if you use spearmint. The oil consists of about 50% menthol, but I couldn't find references for the actually recovered amount when using crystalization only (chemists tend to follow up with a boric acid distillation) but let's say you can recover half of the contained menthol, this would be 15-25 grams of pure menthol crystals.

If your children want just a few portions of ice cream, you should be able to downscale this to 1/10 or less, and get away with a much smaller still. But you'll still need several square meters worth of mint plants, and your children will have to wait a few weeks for their ice cream.

One option would be to make a syrup.

Heating water and sugar with the mint, and straining the mint leaves out would be enough to make a simple syrup - altering the ratio of sugar to water will control thickness and shelf life, altering the mount of mint will change intensity.

Such a syrup can be used as-is on ice cream, flavored syrups often are, but if you're set on a powder the syrup can also be dehydrated through a candy-making process (like rock candy or like hard candy) to give a minty crystalline powder, with a texture somewhere between crushed hard candy and sugar crystals. It'll absorb water and become sticky easily, maybe a silica gel pack or something like would help keep it dry.

Another option would be to skip the syrup step and go directly for mint infused sugar - bruise and tear the mint, layer in a jar with plenty of sugar, stir occasionally. The mint oils will rub off into the sugar crystals, and the spent leaves can be fairly easily sifted from the sugar for use. This is simpler and will give a fairly dry result, but it will take more mint leaves for the same amount of sugar, and extract less of the oils (only what seeps from the surface) - the mint leaves will still hold a lot of their flavor.

A third option would be to try making the powder from dehydrating a tea, without the sugar. This is not an easy option, though the powder it would produce would be strong and purer than the sugared options. You would want to make a strongly brewed tea, reduce it as much as possible, and then dehydrate it down to powder. The extended heating may change or alter flavors, and the amount of powder reclaimed will be very less - enough less that it may be physically difficult to dislodge from the surface it was dehydrated on, as it would tend to form a very thin residue unless the amount of tea (and thus volume of the mint residues) is very large.

Please be aware that these options will make a mint powder, or a mint/sugar powder, which may have different flavors or textures than the menthol powder you're used to.