você pode especificar como
ORDER BY FIELD(MONTH,'January','February','March',...)
SELECT leave_balance.balance, MonthName(leave_balance.date_added) AS month
FROM leave_balance WHERE leave_balance.staff_id_staff = $iid
GROUP BY month, leave_balance.leave_type_id_leave_type
HAVING leave_balance.leave_type_id_leave_type = $leaveBalTypID
ORDER BY FIELD(MONTH,'January','February','March',...,'December');