UK to Nairobi, what to bring?

On putting items in the car it depends on who you ship with, but if they allow it you have to put it on the bill of loading.
I would also not put any Items you really did not want to loose
I am shipping my car with the boot filled

You usually can't put anything in the vehicle if you ship it via roll on roll off (not even spares for the vehicle), but no problem at all if you use a container.  We have shipped a vehicle plus household effects on two occasions, using 40' containers.  The first time we ordered a packing plus door to door service and this cost around £4500 in 2004.  The second time we shipped we dropped the vehicle and effects at the shippers and collected it at Mombasa ourselves - this cost approx £2000 in early 2010.

I am unsure whether you would need to pay duty - you need to check the KRA website.  I do know that for those who have a duty free allowance, personal effects must not be new.  Concerning the vehicle, duty is likely to be payable unless you are working for the UN, or consulate, for example.

With regards to what to ship; furniture may be useful. White goods are easy to buy, as are TV's.  You should consider what you doing in the long term and whether you may have to ship items back again at the end of your contract.