This is the boring article that saves people the most money, so please read it before you buy a policy rather than after.
The altitude clause
Most standard travel policies cover trekking up to a stated height, and that height is very often somewhere between 2,500 and 4,500 metres. Uhuru Peak is 5,895. If your policy stops below that, you're uninsured for the part of the trip where something might actually happen. Look for a policy that names trekking to 6,000 metres, and check that it says trekking rather than mountaineering, since the mountaineering version usually costs more and you don't need it.
Evacuation and repatriation
The cover that matters. You want emergency medical treatment, mountain evacuation and repatriation home, with a figure attached that isn't token. Read how the insurer handles payment too: some pay the hospital directly, others expect you to pay and claim it back, which is a different kind of problem at short notice.
About the helicopter
Helicopter evacuation from Kilimanjaro exists, it's run by private operators, and it's expensive. It's also weather- and daylight-dependent and it can't land everywhere. Plenty of evacuations are done the old way: on foot, on a one-wheeled stretcher, downhill, fast. Don't buy a policy on the assumption that a helicopter will always be the answer, and do check whether yours covers the operator the park actually uses.
Cancellation, and the other half of the policy
Climbs are usually booked months ahead and paid for in advance. Cover for cancellation, curtailment, delayed flights and lost baggage is worth having, and lost baggage is not a theoretical risk here. Bags do go missing on the way to Kilimanjaro International, which is why experienced climbers carry boots and a jacket in hand luggage.
Declare what you've got
Pre-existing conditions have to be declared, including the ones you consider minor. Altitude puts stress on your heart and lungs, and an undeclared condition is the easiest claim in the world for an insurer to refuse.
Carry the details with you
Policy number, the emergency assistance phone number and your insurer's name, printed on paper in your duffel and a photograph on your phone. At 4,600 metres in the dark, nobody is logging into an email account to find a reference number.
One more thing
Ask your operator whether their crew are insured as well. The answer tells you something about the company that no brochure will.
Buy it when you book, not when you pack
Cancellation cover only works if the policy exists before the thing you need to cancel for. Climbs are usually paid months ahead, so buy the policy the week you pay the deposit. Then read the altitude clause before the cooling-off period ends, which is the only time you can cheaply change your mind.