What is the *exact* consistency strength of $MK$? It's well known that the existence of an inaccessible cardinal proves $Con(MK)$. Joel Hamkins has a nice blog post (http://jdh.hamkins.org/km-implies-conzfc/) that explains what you get out of $MK$, and in particular that $MK \vdash Con_n(ZFC)$ for every $n$ [here $Con_n(ZFC)$ is the iteration of the consistency sentence $n$-many times, e.g. $Con_4(ZFC) =_{df} Con(Con(Con(Con(ZFC))))$].
Question: Is the exact consistency strength of $MK$ known?
 A: Thanks for your kind words about my blog post. Let me try to answer your question. 
To describe the consistency strength of a theory or assertion, we
should compare the consistency of that theory or assertion to that
of other more familiar or landmark theories or assertions. For
example, the consistency strength of ZFC plus the continuum
hypothesis is ZFC itself; the consistency strength of ZF+DC+all
sets are Lebesgue measurable is the same as ZFC + there is an
inaccessible cardinal. We compare our given theory or statement to
a landmark theory.
The issue with your question, then, is that KM is itself such
a landmark theory. The exact consistency strength of KM is: KM
itself. Your question is a little like asking, "What is
the exact consistency strength of ZFC plus an inaccessible
cardinal?" The answer would be: ZFC plus an inaccessible cardinal.
But naturally such an answer will not satisfy. Perhaps you could explain what kind of answer you were seeking?
Meanwhile, it is possible to explain how the strength of KM
relates to other large cardinals. It doesn't line up exactly with any of the usual large cardinals. (Although I view KM itself as a kind of large cardinal axiom.) 
Lower bounds. The post on my blog to which you link explains
that KM is strictly stronger than ZFC in consistency strength, and
the argument given there shows that KM implies that there is a
proper class club of cardinals $\kappa$ with $V_\kappa\prec V$.
Thus, these are all worldly cardinals, and by elementarity each of
them will be a limit of worldly cardinals. So one can begin to
climb the degrees of worldliness, in the style of hyperMahloness,
and see that there is a stationary proper class of hyperworldly
cardinals, hyper-hyper worldly cardinals, and so on.
Another lower bound is provided by my recent paper V. Gitman, J.
D. Hamkins, Open determinacy for class games, in review. Namely,
we prove that the principle of clopen determinacy for proper class
games is equivalent over GBC to the principle ETR of elementary
transfinite recursion, which allows transfinite recursion over
proper class well-founded relations, which are not necessarily set
theory. That principle also gives the truth predicate, which leads
to the worldly cardinals as in the previous paragraph. These
theories are strictly weaker than GBC + $\Pi^1_1$-comprehension,
which is also strictly weaker than KM.
Upper bound. Meanwhile, KM is strictly weaker than ZFC + there
is an inaccessible cardinal. This is simply because if $\kappa$ is
an inaccessible cardinal, then $\langle
V_\kappa,\in,V_{\kappa+1}\rangle$ is a model of KM, and so from an
inaccessible cardinal we can deduce $\text{Con}(KM)$ and
$\text{Con}(KM+\text{Con}(KM))$ and more, iterating many times.
Equiconsistencies. It turns out that KM is equiconsistent with
a natural strengthening of KM denoted by $\text{KM}^+$, which
includes the class-choice principle: $\forall x\exists X\
\varphi(x,X)\to\exists Y\forall x\ \varphi(x,Y_x)$. The axiom says
that if for every set $x$ there is a class $X$ with a certain
property, then you can find a class $Y\subset V\times V$, whose
slices $Y_x$ serve as witnesses. Gitman, Johnstone and I have
proved that this assertion is not provable in KM itself, but one
can construct a model of the stronger theory from any model of KM,
so they are equiconsistent.
