I don't recognise the plant, but the infestation sounds like fungus gnats, which live in the dead vegetable matter in the soil around the plant. If it is a well-established infestation, there will be eggs and larvae in the soil, which replace the adults in a few days. You have to keep repeating the treatment until there are none left.
I have seen various cures, but it may take a while. Scrape off as much surface compost as you can and replace it. Spray the leaves with a bug-killer for indoor plants so that the plant itself draws it into its system and does its bit. Feed and water the plant so that it gets stronger, and don't let it dry out again - they prefer dry compost.
Put some left-over wine (drink the rest!) into a shallow dish and stand it nearby. The adults will go there to drink and drown. I have tried this and caught lots of them, and didn't see any more for some weeks, so it seems to work - but I repeated it as soon as I saw another.
I have also seen it recommended that you put a cut slice of raw potato on top of the compost, and the grubs creep up into it. The potato can be thrown away and a new slice laid down. I didn't find this worked, but maybe mine were fussy eaters.