Diagonalization shows that K is not computable. It is easy to show that K is partially computable. So
There exists a problem that is partially computable but not computable.
Although diagonalizations can be carried out for complexity classes, it is not sensitive enough to prove that there is a problem in NP that is not in P.
We do not even know how to prove that there is a problem in PSPACE that is not in P.
That forces us to resort to conjectures. The following are some standard conjectures.
P ⊂ NP ⊂ PSPACE ⊂ EXPTIME
P ⊂ CoNP ⊂ PSPACE ⊂ EXPTIME
NP ≠ CoNP, and neither NP ⊆ CoNP nor CoNP ⊆ NP is true.
You should be aware of the standard conjectures.