I computed 1,568 using a recursive function written in Python. The code is below. N(x,y,n) is the number of ways a king can get from (0,0) to (x,y) in exactly n moves. The idea is that in order to get to (x,y) in n moves, the king must move to an adjacent square in n-1 moves.

Code:

def N(x, y, n):
if x == 0 and y == 0 and n == 0:
return 1
if x > n or y > n:
return 0
if x < 0 or y < 0:
return 0
if x > 7 or y > 7:
return 0
sum = 0
for i in [-1, 0, 1]:
for j in [-1, 0, 1]:
if i != 0 or j != 0:
sum = sum + N(x+i, y+j, n-1)
return sum
print N(0, 7, 8)