There are no direct flights from the US to Nairobi or Mombassa, Kenya. ..Nairobi is main port of entry for Kenya for flights from Europe and Asia. Charters can enter to Mombasa.
Worked at a US Embassy in East Africa 4 years.
A visa is required for US citizens.
The easiest and cheapest way is fly to Nairobi, get passport stamped then get a hop to Mombassa.
Have you never heard of a “chartered flight”?
IIRC David found a BAW flight that went from Mombasa to Glascow to Vancouver (in 61). They allow students, missionary and military to fly free or at reduced rates if the flight wasn’t full.
The rail line build by the British during WWII between Mombasa and Nairobe was/is a primary form of transportation (although large vans/taxis were also used to transport people). It’s depot for the train is next to the the airport in Mombassa that was the main military access at the time. The Coastal Hospital is nearby.