It definitely does not converge. There is a nonintegrable singularity at -1 + 2 Sqrt[2]. It is a pole of order 3, hence not integrable. The fact that increasing MaxRecursion gives ever more large results is an indication of such a singularity. And NIntegrate does issue a warning message to the effect that there might be such a problem.
MaxRecursion
NIntegrate
If I change NIntegrate to Integrate, I get a msg that the Integral does not converge.
After cleaning out the cache, I re calculated, and see a pole at {{k -> 1.82843},, which means the integral won't converge. Sorry about the earlier (hasty, poor) post.
{{k -> 1.82843},
