I get root aphids every time I buy plants from a source I won't name - particularly hybrid primulas
What I have found is that if I leave them in a bowl of Jeys Fluid mix for about 12 hours they seem to die
I used to grow a lot of auriculas
For the brave
Hydrogen Peroxide can be used as a first step before you invest any money in a solution. Hydrogen peroxide works at killing aphid larvae.
Hydrogen peroxide can harm your roots if used to much.
Dry out the soil for a couple of days, then mix 1 part hydrogen peroxide to 4 parts water and soak your infected plants. When the growing medium starts to foam, it means the larvae are now being killed.
for the not so brave
Neem Oil is an excellent natural remedy that works on all type of insects and fungal outbreaks. Make sure the soil is completely dry.
Simply mix two tablespoons per gallon of water and soak the roots of your plants.
Nuclear options
Botanigard ES is something that is quite expensive, but it does the job. This stuff is nasty to the aphid it fills the larvae with a fungus and kills it. If you need the final solution then this is your ticket.
Azamax is a great second alternative after natural remedies; it is 100 percent vegetable base solution that is made for all insects chewing and sucking. Can be used in foliar sprays or when watering plants.
Azatrol a food grade insecticide that works ideal for hydroponic tanks and is perfect for soaking the soil when you have an outbreak.
Azatrol kills the larvae of root aphids.