A simple attack on DESX in time $2^{120}$
It is vulnerable to a sort of meet-in-the-middle attack since you don't really have to brute force $k_1$.  Given a plaintext-ciphertext pair, $P$ and $C$, you can calculate $C' = E(k_2', P \oplus k_3')$ for candidate values $k_2'$ and $k_3'$.  Then, you can determine $k_1' = C' \oplus C$.  So, you only need to brute force the two keys $k_2$ and $k_3$, making the complexity $2^{|\mathcal{K}\cdot\mathcal{M}|}$.

Equivalently, you could attack it from the other side by trying all possible values for $k_1$ and $k_2$, decrypting $C$, and using XOR with the plaintext to determine $k_3$.