You forced me to consult an endgame book for the first time in years, and only then did I learn that this endgame is a draw, since White should always have a perpetual check.

56. Ke5 (best) Qg5+ (now m23 - ...Re3+ is m18)
57. Kd4 (best) Qf6+ (m22)
58. Ke4 (best) Qf3+ (now m23 - ...Qg6+ is m21)
59. Kd4 (best) Qf4+? (now a draw - ...Qf6 is m22)

Then comes a double blunder!
60. Kd5? (allowing m8! - needed Kc5!) Ra5+? (now a draw again - needed ...Rd3 for m8)

And the game remained a draw the rest of the way.

BTW, Black's win after 60...Rd3+ (diagram) might have gone like this:

click for larger view

if 61. Ke6: ...Rd6+ 62. Ke7 Qe5+ 63. Kf7 Rf6+ 64. Kg7 Qg5+ 65. Kh7 Rh6# -or
if 61. Kc5: ...Qd6! 62. Kb5 Rb3+ 64. Ka5 Qb6+ 65. Ka4 Qb5#

As you say, it's a very tricky position for humans. I was hoping to find something instructive from the tablebase.

It appears to suggest that the correct general plan for the attacker (black in this case) is to is to separate the defending king from its Q+R, after which the Q+R can force mate astonishingly fast.

So, defensively, it makes sense to move your king and Q+R closer. Perhaps that idea might have helped you avoid the two potential losing moves 55. Ke6? and 60. Kd5? (in both cases, Kc5! was preferred).