I think that this is not
mate in 5
but
mate in 6
With the first move being
1. Kd3
Analyzing all of black options:
After this move, black cannot play
1.b5 or 1.Ne8
bacause in these cases
black has no way to stop
2. Ke4 with the idea 3.Kf3 and 4.Bxg2#
One possibility for black is the move
1. Nh5
In this case, white cannot play
2.Ke4 because of 2.Ng3 negates every checkmate in 5 or 6: 3.Kf3 Nxf1 or 3.Nxg3 Kxg3)
but has the option to play
2.Rxh5 forcing mate in 6 after 2. b5 3.e8=Q b4 4.Rb5 Bg3 5.Qh8+ Bh4 6.Rf3#
2.e8=Q which is also mate in 6, forcing 2.Nf4+ (because of 2.b5 3.Qxh5 b4 4. Qxh4#) and forcing white to move the king somewhere, having ideas of checkmating with the queen in 2 on Qxg4# or Qxh4#. However, black does always have the option to give check at least once in all of these lines leading to a variety of forced mates in 6
Looking at the other possible first moves for black:
1.Ne6
is also mate in 6: It leads to 2.Bxe6 b5 3.e8=Q b4 4.d5. Here, black has three options,
4.Nxe5+ 5.Qxe5 Kg4 6.Be6# or
4.Nf6 5.Qe6+ g4 6.Bxg2# or
4.Nh6 5.Qe6+ g4 6.Bxg2# (same as before)
Leaving just
1.Nf5
which is also mate in 6 after
2.Bxf5 e5 3.e8=Q e4 4.Be4 and black having the options
4.Nxe5+ 5.Qxe5 Kg4 6.Qf5# or
black moving the knight to either f6 or h6 which is followed by
5.Qe6+ Ng4 6.Bxg2#