I dunno, that sounds a bit too fishy.

If there weren't such a constant, I think it would follow that in effect K.C. wouldn't be generally incomputable. (I may dwell on it further, once I'm sober in the morrow.)

(...) looking for one that has greater than a certain complexity.

The problem with your approach is that K. C. isn't computable, so you wouldn't know if you've found the exact K.C. of that bit string. Even iterating through all programs that generate it wouldn't give you the answer to that one, since you're not iterating from lowest K.C. to highest, but only from the uncompressed smallest program upwards.