Defintely could be improved. I drilled my kids on KR v K and KQ v K and insisted they do it at near perfection.
If you are retreating your King, something is wrong. If you are pushing the King to the center, you are doing something wrong. Push to edge then mate is simple.
Move 56, Why not Rf5 pushing him toward the 8th-rank instead of your pushing him up to the 1st rank?
Move 62, play Kd4
Move 64, play Rc4
Please review http://www.chess.com/article/view/basic-checkmates-king-and-rook-mate
Once you are more advanced, practice with an endgame tablebase here: http://www.k4it.de/index.php?topic=egtb&lang=en
Hope that helps.
What's the best way to win such an endgame? Could I have done this more efficiently?
Thanks