how do black mate in 4?

playing as black and during the late game, i think i missed many checkmate opportunities. the analysis said i would mate in 4 on move 54.
can you help me how to best mate as black? i ended up winning the game but i somehow think there might be better more effective move to win the game

the position at the end of move 54:
white: pawn a2, Kg3
black: Kb6, pawn g7, Qh1+, Rh5, Kh8

(sorry i cant upload picture here)

all the moves:
1. e4 e5
2. Bc4 Nf6
3. d3 Ng4
4. Qxg4 d5
5. exd5 Bxg4
6. h3 Bd7
7. Nf3 Bb4+
8. c3 Ba5
9. b4 Bb6
10. Bg5 f6
11. Be3 Bxe3
12. fxe3 O-O
13. d6+ Kh8
14. Bd5 cxd6
15. Bxb7 Bc6
16. Bxc6 Nxc6
17. b5 Nb8
18. Na3 f5
19. e4 d5
20. Nxe5 fxe4
21. dxe4 dxe4
22. Rf1 Nd7
23. Rxf8+ Nxf8
24. Nf7+ Kg8
25. Nxd8 Rxd8
26. Rd1 Re8
27. c4 e3
28. Rd3 Ng6
29. Ke2 Nf4+
30. Kf3 Nxd3
31. g4 Ne1+
32. Ke2 Ng2
33. g5 Nf4+
34. Kf3 Nxh3
35. Nc2 e2
36. Kq2 Nf4+
37. Kf2 h6
38. g6 h5
39. Nd4 e1=Q+
40. Kf3 Qt2+
41. Kxt2 h4
42. Kt3 Rf8
43. Nc6 h3
44. Ne7+ Kh8
45. Kg3 Rf6
46. c5 Nxg6
47. Nd5 Rf5
48. Ne3 Rh5
49. Nc4 h2
50. c6 Ne7
51. c7 h1=Q
52. c8=Q+ Nxc8
53. b6 axb6
54. Nxb6 Nxb6
55. a4 Nxa4
56. Kf2 Rf5+
57. Kg3 Rg5+
58. Kf4 Qh4+
59. Kf3 Qg4+
60. Ke3 Re5+
61. Kd3 Qe4+
62. Kd2 Rd5+
63. Kc1 Qe1+
64. Kc2 Qc3+
65. Kb1 Rd1+
66. Ka2 Qb2#

thanks