Does having a bunch of plaintext/ciphertext pairs help you break a ciphertext with unknown plaintext? Let say we have $N$ ciphertexts and each of them looks like this:

2iob3gRLvabM0QnMtgtMG5y4kCm=2xmLkasBUasLkQkjyQtQ3NFhU5UNkQ95U1nmk5UltxwZt1DltaojkQuHkxyXQI7=GYzYYf7t25ymQd7wtHJ52xWM3NJR3QTLUxmI3IZMU54NkitMGHz=2xmLvQnVtQUBk5Wa

assume it's encrypted URL string and we know output (actual URL) of each $N$ ciphertexts. In addition we know that the key is $m$ digits. So my question is how to assess complexity of breaking the $N+1$ ciphertext?
The known plaintext is only a useful tool for establishing that you actually found the right key if we assume a secure cipher (and mode of operation, etc.). It won't help you creating a faster than brute-force attack, as ciphers are assumed to be secure against known plaintext attacks.

So the complexity of breaking the cipher is simply still $m - 1$ (half the key space) on average, i.e. no change from normal brute forcing.