The data was coded wrongly mixing the false positive and false negative and so the true values, and it's impossible to get the detailed data. Can I do cohen's Kappa test?
Cohn's weighted kappa value works when attempting to measure accordance/discordance, yes, but no idea what this will tell you as you won't know what's right and wrong.
I think, the answer strongly depends upon the formulation of the problem that you need to solve (and that you haven't described here).
Do I get it right that you have effectively a 2*2 contingency table of some binary ("yes/no") test results against the real existence/nonexistence of a feature (and instead of two off-diagonal elements of the table you know their sum only)? And what do you want to know about the test (and what to use it for)?
I think you can only test the equality of the proportion of positives obtained with the two tests, but not the agreement.
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