How to get rid of tiny insects in soil of potted plants?

I would sprinkle either diatomaceous earth on top of the soil, or borax.

You will need to repot the plants with new soil and get rid of the old soil, gently rinse the plants and roots to make sure they are all gone. Clean the pots with bleach water and spray with vinegar. Let it sit on it a few minutes and rinse it off. Repot with the new soil. They are procreating in the soil, you must have some kind of fungal growth in the soil. They are also attracted to the moisture in the traps in your sink. Run extremely hot of boiling water down your drains to take care of them. Get an attractant to get the strays that are hanging around. If there are still a lot around, you can use fly strips to get rid of them, too, they may not be nice to look at, but they do work until you get rid of the gnats.

Spray just the soil with Simple Green. Spray all your drains in the house with it too. It is a non toxic disinfectant that will kill small insects. Keep spraying every day until you see no more, then use once a week as needed in the drains only. It will prevent build up.

I googled "what is the name for the bug called no see ums and got this:

"They are called midgies in Scotland, are also known as no-see-ums, sand flies, punkies, and others in North America, and sandflies in Australia. (The name "sandfly" is ambiguous, as it is also applied informally to many other flies, such as the subfamily Phlebotominae.)"

I then looked for something to repel them and got this:

https://nonoseeum.com/