Since this is the entire deck, there are exactly 1080 sets present.
As far as how many can be easily "spotted" in the magic hypercube...
Within each 3x3 square, each of the 3 rows, 3 columns, and 2 diagonals are sets; also each corner forms a set with the middle card of each of the opposite sides of the square. That's a total of 12 sets per 3x3 square. This kind of 3x3 square is called a "magic square".
So 9 x 12 = 108 sets so far.
Each column and each row of the overall picture also makes up a magic square, which is 12 sets, though 3 of the 12 will have already been counted in the previous step. So 2 x 9 x 9 = 162 more sets, 270 so far.
Now choose a position like "upper left", and look at that card from each of the 9 3x3 squares. Now you have another magic square with 12 sets, 6 of which we already counted in the previous step, so we have 9 x 6 = 54 more sets for a total so far of 324.
Now pick any card in the imagine (except the three solid red squiggles in the center). Find the card that is in the opposite position to this card (as if you turned your screen upside down then looked in the same position). Those two cards together with the three solid red squiggles card form -- oh you'll never guess.
