I travel to Kenya a fair bit and usually go with friends and their DC (now 4 but been going since 6 months) You'll be there in the dry season (dec-march). I probably wouldn't go in the wet season, mainly because it's miserable weather!
Nairobi, the immediate surroundings and the highlands (except valleys) are low risk for malaria but you do need to take precautions particularly at twilight times. Nets, deet, long sleeves etc.
In terms of yellow fever, you only need to produce a yellow fever certificate if you are coming from a country affected by yellow fever, so if you fly direct from UK it should be ok. Consider what your yellow fever exposure is likely to be.
The CDC guidance is American but very comprehensive and is what I tend to look up, along with WHO guidance. it doesn't recommend yellow fever vaccination if your stay is limited to the North East/Kilifi/Kwale some other provinces plus Mombasa and Nairobi
Given your description of where you'll be staying, I imagine this will be air-conditioned, sealed compound type areas without open windows and stagnant water lying around.
I'm a doctor and often find that some poorly-travelled doctors do have quite old-fashioned ideas about what cities will be like in developing countries. This may be related to the fact that a lot of our tropical diseases education is based around diseases that are more endemic in rural and impoverished areas. For example the GP thinking you'd be walking to the market - in Nairobi I'd expect you to be driving in an air conditioned car to a shopping mall! (NB I recommend the Hub - good food court with lots of international variety if you get sick of Kenyan food and KFC/Dominos if you want fast food, good range of shops, a handy supermarket and some kids activities. They've also got super fast free wifi.)
Overall, I'd have a think about exactly where you'll be visiting - get on google maps (streetview isn't everywhere but there are lots of pics) - to get a sense of the areas and plan from there.