Could this kind of inaccurate sacrifice be countered? I recently played a Lichess Bullet Game as White.

[FEN ""]

1.e4 e5 2.h4 Nc6 3.d3 Nf6 4.g3 Bc5 5.Nc3 d6 6.a3 O-O 7.b4 Bxf2+ 8.Kxf2 Ng4+ 9.Ke1 a6 10.Bh3 Nd4 11.Bxg4 b5 12.Bxc8 Rxc8 13.Nge2 Nxe2 14.Qxe2 c5 15.bxc5 dxc5 16.Rb1 c4 17.dxc4 bxc4 18.Be3 h6 19.g4 f6 20.Bxh6 gxh6 21.Qe3 Kf7 22.Qxh6 Ke7  23.Rb7+ Ke6 24.Rf1 Rh8 25.Rxf6+ Qxf6 26.Re7+ Kxe7 27. Nd5+ Kf7 28.Qxf6+ Kg8 29.Ne7+ Kh7 30.Qg6#


After Black's 24th move, I thought I was doomed, but I found the rook sacrifice, Rxf6, which forces Black to capture Qxf6. I could have gone for Rb6+ next, taking their queen, but I missed that and played Re7+ sacrificing both rooks. They had to capture with the king, and now, what I foreseen, is that Nd5+ forks the Black king and the queen. 

My question is this: If Black ran better with his King, would they have a chance to defend against my attack? It seems to me that d7 is not a better square for black to move his king to because he has to block Qe7+ and there is also the Nb6+ threat of forking the rook. It's possible that their next best move would be Rce8, which is met with 1. Qf7+ Kd6 2. Qc7+ Ke6 3. Qc6+ Kf7 4. Qf6+ Kf8 Ne7+, forcing him to take with the rook, because otherwise it's mate. It looks like the game was mine anyways.

Nevertheless makes for a more spectacular win.
When asking questions about tactics and sacrifices, chess engines are usually the most efficient and accurate way of determining the answer. The following is based on analysis from Houdini 6.03 in tactical mode:

By the end of move 24, in the following position, a forced mate (in 8) already exists for white

[FEN "2rq3r/1R6/p3kp1Q/4p3/2p1P1PP/P1N5/2P5/4KR2 w - - 5 25"]

1.Rxf6+ { #8/58 } 1...Qxf6 2.Rb6+ Rc6 3.Rxc6+ Kd7 4.Qxf6 Rh6 5.Qxh6 Ke7 
6.Rc7+ Kd8 7.Qd6+ Ke8 8.Qe7#


And Rxf6+ was indeed the optimal move. Later on, Rb6+ was completely fine, but not optimal (see analysis above). As it turns out, you lose your mate in 7 and are tragically left with the following mate in 19:

[FEN "2r4r/4R3/p3kq1Q/4p3/2p1P1PP/P1N5/2P5/4K3 b - - 1 26"]

1...Kxe7 { #19/50 } 2.Nd5+ Kd7 3.Qxf6 Rhf8 4.Qe7+ Kc6 5.Nb4+ Kb5 6.Qxe5+ Kb6 7.Qd6+ Kb7 8.Qxa6+ Kc7 9.Qc6+ Kb8 10.Qb6+ Ka8 11.Nc6 Rxc6 12.Qxc6+ Kb8 13.Qd6+ Kc8 14.Qxf8+ Kd7 15.e5 Ke6 16.Qd6+ Kf7 17.Qf6+ Kg8 18.h5 Kh7 19.h6 Kg8 20.Qg7#


It turns out the best move for black (the one which holds off mate for longest) is in fact to take your rook and allow themselves to be forked. 

On move 27, the played move Kf7 allows the mate in 3. The best option, Kd7 is shown in the mating line above. It delays mate much longer than 3 moves, but obviously still loses.