Here’s a map, for anyone interested. Obama’s home village, and where his grandmother lives, is on the northern edge of Lake Victoria, sort of near that village of “Kisumu” on the map, though a little west of it. Nairobi, the capital city with many hospitals, is in the center of the map, near the “A109” designation. Mombasa is on the far right of the map, right on the coast.
So that makes Mombasa a particularly unlikely place to pick if one were making up a Kenyan birth story—rather than dealing with actual facts.