Kevin has answered your question in comments, but it might help to make some further remarks:
If $p$ is odd and $E$ has good reduction at $p$, then the image of $K_p$ is independent of $E$ (i.e. does not depend on the particular $E$ other then requiring that it has good reduction). To be precise, the image will be $\mathbb Z_p^{\times}/(\mathbb Z_p^{\times})2 \times \mathbb Z_p^{\times}/(\mathbb Z_p^{\times})^2.$
Thus there is not much chance that you will be able to extract any information about the global elliptic curve $E$ from knowing the image of $K_p$. (Even if $p = 2$ and/or the reduction is bad, there is very little information specific to $E$ in the image of $K_p$; it will just depend on generalities about the reduction type of $E$.)