Determine the Key given ciphertexts and plaintexts? Given two sets of plaintexts and their corresponding ciphertexts, how does one determine the key $K$?

The ciphertexts are:

$$C_1= (P_1 \oplus K_0) + K_1$$

and

$$C_2= (P_2 \oplus K_0) + K_1$$

$K = K_0 || K_1$ (string of bits).

where $+$ means addition mod $2^n$.

How do we find the unknowns (eg, $K_1$)?
There is a straightforward brute force method. For example, take the lowest $8$ bits of everything and check for valid values of $K_1$ and $K_2$, mod $2^8$. You will need about $2^{16}$ checks to get the lower $8$ bits of $K_1$ and $K_2$. Proceed then to values mod $2^{16}$, as you know the lower $8$ bits of $K_1$ and $K_2$, only bits $8\dots15$ of these must be checked, the effort is $2^{16}$ again. Proceed until you get $K_1$ and $K_2$ in full.

Better, do the same but instead of $8$ bit increments do $1$ bit increments. So, brute force for values mod 2, mod $4$ ... mod $2^n$. You will need to check $4$ different values each step (one bit from $K_1$ and another from $K_2$), and do this $n$ times. Therefore, the cost of finding $K_1$ and $K_2$ is $4n$, meaning this isn't just polynomial in respect of the number of bits: it's linear.

However, must be noted that if $P_1 \equiv P_2 \pmod{2^i}$, then the first $i$ bits of $K_1$ and $K_2$ are indeterminable, so more plaintext/ciphertext pairs will be needed until finding some pair where $P_1 \not\equiv P_2 \pmod{2}$, to retrieve then the whole keys.

Update: Probably there are (for a given $n$) $2$ pairs $(K_1,K_2)$ compatible with an adequate $(P_1,P_2)$ pair, so this method should check incrementally $2$ pairs of keys.