Easiest way to decide is to backdate.
You need to be at the WAT at 12:15.
It takes 10-20 minutes to walk from the gate to the KS depending on how fast you walk/crowds, etc.
Depending on if you want to be 'safe' with your transport to AK from the Poly, allow 30-60 minutes of bus time.
Depending on how your family likes to eat, budget 45-90 minutes for breakfast.
A "safe" estimate, based on everything going wrong and the meal taking forever, would be 8:45. 10:00 would be the absolute later, but if it were me I would do 9:30. That gives you plenty of time to eat, allowance for missing one bus/one bus being full, and time for a leisurely stroll to the Africa section to let the story sink in.