The largest sum possible for the queen’s journey is 305. It turns out that there are 36 paths that achieve such a score, one of which is:

c3, c7, d8, d1, g4, h3, a3, a4, e8, f7, b3, a3, e7, e5, b5, c4, f4, d4, f2, c2, c8, e8, b5, c4, c2, b3, f7, e8, h8,

which scores as 8 + 29 + 21 + 16 + 21 + 16 + 21 + 5 + 21 + 16 + 21 + 16 + 1 + 9 + 17 + 9 + 8 + 9 + 8 + 9 – 7 + 8 + 9 + 1 + 4 + 6 – 1 + 2 + 2 = 305.