Okay, I think this does it.

In addition to having an embedding of the Iwahori-Hecke algebra into the affine Hecke algebra, $\iota:H_n^{\mathrm{fin}}\hookrightarrow H_n^{\mathrm{aff}}$, there is also a surjective homomorphism $\pi:H_n^{\mathrm{aff}}\twoheadrightarrow H_n^{\mathrm{fin}}$ which is the identity on $H_n^{\mathrm{fin}}\subset H_n^{\mathrm{aff}}$, sends $X_1\mapsto 1=L_1$, and $X_i\mapsto L_i$, for $2\leq i\leq n$, where $$L_i=q^{1-i}T_{i-1}T_{i-2}\cdots T_1T_1\cdots T_{i-2}T_{i-1}$$ is the $i$th Jucys-Murphy element.

Now, the center $Z^{\mathrm{aff}}$ of $H_n^{\mathrm{aff}}$ consists of symmetric Laurent polynomials in the $X_i$, while the center $Z^{\mathrm{fin}}$ of $H_n^{\mathrm{fin}}$ consists of symmetric Laurent polynomials in the Jucys-Murphy elements.

Evidently, the centralizer of $H_n^{\mathrm{fin}}$ in $H_n^{\mathrm{aff}}$ is $\pi^{-1}(Z^{\mathrm{fin}})$. So, the centralizer is all symmetric Laurent polynomials in the $X_i$ and $L_i$ or are symmetric after exchanging some $X_i^{\pm1}\leftrightarrow L_i^{\pm1}$.

This is not a complete answer, but does give some information.

In addition to having an embedding of the Iwahori-Hecke algebra into the affine Hecke algebra, $\iota:H_n^{\mathrm{fin}}\hookrightarrow H_n^{\mathrm{aff}}$, there is also a surjective homomorphism $\pi:H_n^{\mathrm{aff}}\twoheadrightarrow H_n^{\mathrm{fin}}$ which is the identity on $H_n^{\mathrm{fin}}\subset H_n^{\mathrm{aff}}$, sends $X_1\mapsto 1=L_1$, and $X_i\mapsto L_i$, for $2\leq i\leq n$, where $$L_i=q^{1-i}T_{i-1}T_{i-2}\cdots T_1T_1\cdots T_{i-2}T_{i-1}$$ is the $i$th Jucys-Murphy element.

Now, the center $Z^{\mathrm{aff}}$ of $H_n^{\mathrm{aff}}$ consists of symmetric Laurent polynomials in the $X_i$, while the center $Z^{\mathrm{fin}}$ of $H_n^{\mathrm{fin}}$ consists of symmetric Laurent polynomials in the Jucys-Murphy elements.

Evidently, the centralizer of $H_n^{\mathrm{fin}}$ in $H_n^{\mathrm{aff}}$ is contained in $\pi^{-1}(Z^{\mathrm{fin}})$.

Okay, I think this does it.

In addition to having an embedding of the Iwahori-Hecke algebra into the affine Hecke algebra, $\iota:H_n^{\mathrm{fin}}\hookrightarrow H_n^{\mathrm{aff}}$, there is also a surjective homomorphism $\pi:H_n^{\mathrm{aff}}\twoheadrightarrow H_n^{\mathrm{fin}}$ which is the identity on $H_n^{\mathrm{fin}}\subset H_n^{\mathrm{aff}}$, sends $X_1\mapsto 1$, and $X_i\mapsto L_i$, for $2\leq i\leq n$, where $$L_i=q^{1-i}T_{i-1}T_{i-2}\cdots T_1T_1\cdots T_{i-2}T_{i-1}$$ is the $i$th Jucys-Murphy element.

Now, the center $Z^{\mathrm{aff}}$ of $H_n^{\mathrm{aff}}$ consists of symmetric Laurent polynomials in the $X_i$, while the center $Z^{\mathrm{fin}}$ of $H_n^{\mathrm{fin}}$ consists of symmetric Laurent polynomials in the Jucys-Murphy elements.

Evidently, the centralizer of $H_n^{\mathrm{fin}}$ in $H_n^{\mathrm{aff}}$ is $\pi^{-1}(Z^{\mathrm{fin}})$. From the definition of $\pi$, it is clear that this centralizer is generated by $Z^{\mathrm{aff}}$ and $Z^{\mathrm{fin}}$.

Okay, I think this does it.

In addition to having an embedding of the Iwahori-Hecke algebra into the affine Hecke algebra, $\iota:H_n^{\mathrm{fin}}\hookrightarrow H_n^{\mathrm{aff}}$, there is also a surjective homomorphism $\pi:H_n^{\mathrm{aff}}\twoheadrightarrow H_n^{\mathrm{fin}}$ which is the identity on $H_n^{\mathrm{fin}}\subset H_n^{\mathrm{aff}}$, sends $X_1\mapsto 1=L_1$, and $X_i\mapsto L_i$, for $2\leq i\leq n$, where $$L_i=q^{1-i}T_{i-1}T_{i-2}\cdots T_1T_1\cdots T_{i-2}T_{i-1}$$ is the $i$th Jucys-Murphy element.

Now, the center $Z^{\mathrm{aff}}$ of $H_n^{\mathrm{aff}}$ consists of symmetric Laurent polynomials in the $X_i$, while the center $Z^{\mathrm{fin}}$ of $H_n^{\mathrm{fin}}$ consists of symmetric Laurent polynomials in the Jucys-Murphy elements.

Evidently, the centralizer of $H_n^{\mathrm{fin}}$ in $H_n^{\mathrm{aff}}$ is $\pi^{-1}(Z^{\mathrm{fin}})$. So, the centralizer is all symmetric Laurent polynomials in the $X_i$ and $L_i$ or are symmetric after exchanging some $X_i^{\pm1}\leftrightarrow L_i^{\pm1}$.