I think we can at least set a maximum for the number of knights needed.
First off, the special cases. Checkmate is impossible on a 3x3 (or smaller) board. The knights cannot control the middle square and can thus never give checkmate. On a 4x4 board, the board is so small that if there are 4 knights and the king starts in the middle, the king is next to one of the knights and may be able to take it immediately and fork the remaining 3 for an easy draw, or if you move the king to the corner so it doesn't start in check, it starts the game immediately stalemated.
In the following diagram, the black pawns represent squares that the king cannot cross:
[FEN "3k4/8/8/ppp2ppp/2pppp2/NN4NN/8/8 w - - 0 1"]
If you imagine that pattern continuing, each pair of knights is controlling 6 files. So, to make a wall covering the entire board, you would need 2 knights for every 6 files. An additional 2 knights could ensure that the wall can slowly advance with no gaps. The wall can continue to advance until the opposing king is pinned to the back rank, and after that checkmate should be easy. So, on an NxN board of size at least 5, a maximum of
ceiling(N/6) * 2 + 2 knights should be enough to checkmate. On a 300x300 board, which contains 90,000 squares, 102 knights would therefore be sufficient. (This implies a maximum of 6 knights needed on a regulation 8x8 board, which I'm pretty sure is more than what is actually needed - 4 seems like it would be enough.)
There are probably better methods which could reduce this number, but I give it as a starting point.
Are they fast enough to surround opponent's king in very limited constant count for various finite board sizes
It could never be constant (and that part won't change even if the side with the knights also has a king.) After all, only 2 knights (or a king) can chase the king, and 2 knights (or a king... or even 2 knights and a king) cannot force checkmate, so at the very least the opposing king could run until it hit the edge of the board, and the amount of moves that takes is a function of the board size. Therefore, no checkmating method could be faster than O(N). The method I give of advancing a wall seems to be O(N^2) since the wall has further to go and takes longer to advance the bigger the board is.