EDIT: This answer in incorrect, Noam solved the problem in the accepted answer.
I'm leaving it as an example of an unsuccessful attempt. My mistake was to try to keep the wK on the 6th rank, where it cannot make tempo moves, and push the bK with the knights. Noam proved that the reverse strategy is the correct one: knights guard the bK, the wK pushes it sideways, and thus the wK has tempo moves between the 7th and the 8th rank.
After the question has been edited, I will try and demonstrate how White fail to mate on e8 starting from the same random position I used before (conveniently, it is not too far from e8).
The main point is that if you use a knight to block one escape square along the edge (say, d8), then you are basically reduced to mate in the hence created 'artificial corner' (here, e8) using a king and two knights. And just as in the KNNK ending, due to impossibility to attack the last black square (f8) and the last light square (e8) with the same knight, White fails and stalemates.
[FEN "6kN/8/4NK2/8/6N1/8/8/8 w - - 0 1"]
1.Nf7 Kh7 2. Nfg5 Kh8 3.Kg6 Kg8 4.Ne5 Kh8 5.Nd8 Kg8 6.Ndc6 Kh8 7.Ngf7 Kg8 8.Kh6 Kf8 (8.Kf6 {the other plan: the king blocks in front and the two knights try to attack g8, then f8, then e8.} Kf8 9.Ng6 Kg8 10.Nge7 Kf8 11.Nfe5 Ke8 12.Ke6 Kf8 {and you cannot force the king to e8 and deliver mate.} 13.N7g6 Ke8 (13...Kg7 {escapes}) 14.Ng4 {and if it was White's turn the next move would be Nf6#, but this is stalemate.}) 9.Kh7 Ke8 10.Nd6 Kf8 11.Nf5 (11.Ng6 {mates on the wrong square}) Ke8 12.Nfd4 (12.Kg7 {stalemate}) Kf8 13.Ne6 Ke8 14.Nc7 {The knight can control either f8 or e8, but not both.}
Of course, if the square you have picked is in a corner, White succeeds. I feel like he would fail on g8, but I leave it for someone else to check.