Part of the house Hlaalu advancement involves recruiting some miners for a mine I inherited. Two candidates were mentioned, one of whom is found in the Tel Aruhn underground.
I talked to him and he agreed to come work for me and I continued on with whatever quest I was on at the time.
A few days later I was back at Tel Aruhn and the guy was still standing there.
This time I noticed a slave trader outside the underground, and when I asked her about "recruit some miners" she said she had none to sell.

So how do I get her to sell me the miner, or does he just do his egg mining through long range telekinesis?

Try Ilasour Tansumiran in Gnisis. If I remember correctly, it takes about 3 days for him to get there.