How many different checkmate positions are there? A Super-Computer problem. Not with any conditions or hypothetical situations, just simply the number of checkmates that can be arrived at in a properly played game.
Since there are 10^umpty possible chess positions (an obvious upper bound is 13^64, and estimates are 10^45+-5), even a supercomputer is useless.
The following "Monte Carlo" attempt is a feasible looking way to give an approximate answer, though:
- Randomly generate a position (already nontrivial: how to generate truely random material in the sense that it's the correct proportion of legal positions?)
- Make a flimsy legality test (material, impossible checks - a total one is very hard!)
- Test whether it's a mate.
- Rinse and repeat.
Assume that you test p positions, m are mate and there is a total t of legal chess positions, your answer is approximately t*m/p. I only fear that m/p is very low (for very full boards, in light positions things are different) and you again need a far too high p to get meaningful stats.
I strongly advise to make first tests on smaller board. For example, 3x4 chess is tablebase-solved and the analog of your value could readily be extracted.
It depends a bit on what you mean by 'checkmate position'. One meaning could be that such positions are defined by active pieces only: in this case a back-row mate with two rooks would count only the rooks, and ignore all pieces that does not contribute (say, by guarding a field twice). Or, it could mean all possible such positions with or without participating own king, and might even make distinctions of mates on the first row, the last row, or either of the edge files, although the fundamental mate would be the same.
In Iuri Akobia's World Anthology of Chess Studies, vol. II: 4492 Studies With Mate (1994), the reference section contains the mate positions that are found in the same volume. They're intended for finding studies in the same volume, not to answer your question. But they may help getting into what the question really is. (Although it is not relevant, there are slightly more than 2000 of them in this volume.)