British colony. I'd bet the UK still has copies of the colonial records.
Furtherermore, assuming for the sake of argument that Obama was born there and that there was a birth certificate documenting it, why would the Brits have necessarily maintained a copy of such birth certificate when Kenya gained independence? It would seem more likely that they merely left that type of record in the hands of the new Kenyan government at the time. And with all the succeeding changes in that Kenyan government and civil strife, chances of such a birth record surviving wouldn't seem all that good.
Please to not think of a possible Kenyan birth certificate in the same terms as you would an American or Canadian or Western European certificate.