SHA1 padding on length extension attack
Yes, the length extension is identical for key sizes that are a multiple of 64 as SHA-1 has an input block size of 512 bits. So you would have to generate the exact same padding for the same message in that case. However, commonly this includes the encoding of the length of the input message of the hash in bits.

The output of a number of blocks of SHA-1 cannot be distinguished from random if the input is sufficiently randomized. This is definitely true if a fully randomized key is used at the start of the hash. So you cannot get any information out of it. This is also true for any MD5, SHA-2 or any other Merkle-Damgård hash without any pre/post-processing to avoid it.

The fact that you can test your attack successfully while the sizes are unknown  in advance must therefore mean that you have some kind of oracle to test against. And this oracle simply returns success if you guessed the total message input size correctly, which is the addition of the key size and message size of the keyed hash.

For most protocols you'd know the key size, or consider it known due to the Kerckhoff principle. This is just a configuration parameter after all, it is not part of the security delivered by the key. However, that doesn't mean that you can calculate it from just the output of the hash function.