Tiny black insect on Kaffir Lime Plant

We bought a Kaffir Lime Plant from a UK Nursery about two years ago. It's lived quite happily in our kitchen ever since until about a month ago.

We first noticed that the leaves had a slightly mottled appearance but upon closer inspection we discovered tiny black insects on the leaves. They are extremely small, so we can't actually see any distinct features but we can see them moving!

Does anyone know what this could be and how on earth we can treat it? The plant seems to be deteriorating each day, despite all attempts with careful watering, feeding, etc.

Here are two photos of (a) the mottled leaves at the top of the plant and (b) a close up of the insects.

 Thanks in advance for any help!