After the comment of Jez, here is the corrected answer:

There is the well-defined vertical bundle $VTM\subset TTM\to TM$ given as the kernel of the (differential of the) projection.
Moreover, for any $v\in TM,$ there is a natural identification $\Phi$ of $V_vTM$ with $T_\pi(v)M.$

Using the Levi-Civita connection (or ans other connection), you can construct a horizontal bundle $HTM\subset TTM\to TM$ with the properties:
(1)
$$HTM\oplus VTM=TTM,$$
$d\pi\colon H_vTM\to T_{\pi(v)}M$ is an isomorphism for any $v\in TM$
and (3)
for any vector fields X and Y we have
$$\Phi^{-1} \nabla_XY= \pi^{VTM} dY(X),$$
where $\pi^{VTM}$ is the projection to $VTM$ corresponding to the splitting (1). The proof is the same as the proof of existence of the geodesic spray. From these properties, you obtain that $K=\Phi\circ \pi^{VTM}$ is satisfying your equation.