The tiny bugs could be soil mites or springtails. They feed on damp organic matter in the soil. Over wet soil helps them thrive.

If it were me, I would take a few good leaves and start new plants. Stick your leaves in your potting mix with perlite. 50/50. Make sure your mix is just moist, not wet. Put in airtight covered container. I use the aluminum pans I get at Walmart. They have clear covers. In a month or two you will have new babies.
Then take out your original plant and wash off roots. Trim away the bad leaves. Plant the violet in your moist light mix and place in baggie. I think your gnats are comming from soil that is too wet. I'm sure others will add more info to help you out, but at least this will be a start to save your violet.

I would remove a couple of lower rows of leaves and repot the plant in the smaller pot. Like half of the current one. Exactly as Ima_digger and Tommyr pointed - heavy wet soil = is a paradise for fungus gnats - mix a new soil - cut it half and half with perlite to lighten it up.
The gnats live in organic soil, if you get vermiculite and when you are repotting - fill the last inch of pot with vermiculite - you will significantly reduce the gnats. I do not think it is possible to get rid of them without pesticide. The adult gnats are flying and laying eggs - and the larva lives in a first inch of soil. You can get some mild pesticide - like neem oil based 3-in-1 spray, and soak the soil with it.

The first thing to do - to keep your pot on a dry side. It will really help with bugs and will stop the decline. The pot is too big, the soil is too wet - and this is the main problem.

Good Luck

Irina