I cannot think about anything in the walking distance from Wilson but on Ngong Road you would be able to find serviced apartment - cleaning and even bed linen change is usually provided yet you have the freedom if having an apartment with your own kitchen and all the bits and pieces.
Unfortunately traffic in Nairobi is a serious issue and unless you stay right at the airport you will have to struggle through traffic. Langata road is very busy, Mbagathi Way (going to Ngong Road) is also busy, so no way out of traffic.
I am sure Flying Doctors have an admin office and that you are not the only ones in similar situation, why don't you ask them, perhaps they have some recommended accommodation which people working with them temporary use for their families?
Once you get here you will be able to contact a few operators and see what they can offer. As you will be based in Nairobi possibly you could do a self drive safari and will need to book accommodation only. Also I am sure Flying Doctors will have their travel agent/tour operator who they can recommend.
Tijara Beach Hotel