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 I don't think that there is 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 $E$.)

Have you looked at the discussion of descent and Selmer groups in (e.g.) Silverman's book? If you do, you'll see that the problem of doing descent involves looking at *every* prime (the point being that every prime intervenes in the definition of the Selmer group).

Concretely,
in the case you are looking at, which I guess is an $E$ all of whose $2$-torsion is defined over $\mathbb Q$, what one sees is that if $P \in E(\mathbb Q)$, and we solve
$P = 2Q$, then $Q$ is defined over a biquadratic extension of $\mathbb Q$ which is unramified away from $2$ and the primes of bad reduction. This greatly limits the possibilities, and is the basis for why descent works. In particular, the weak Mordell--Weil theorem --- i.e. the statement that $E(\mathbb Q)/2 E(\mathbb Q)$ is finite --- in this context is essentially the statement that there are only finitely many biquadratic extensions of $\mathbb Q$ with prescribed ramification.

If you focus on just one (or even a finite number) of $p$, you are throwing away this basic fact (i.e. that the field of definition of $Q$ is unramified away from finitely many primes).