[Phizo]

I thought about doing it the lazy way( just iterating through XORing the remainder of the text w/ the first n bytes ), but I'm unsure if I'm missing something obvious that would make it easier.

You could do that, but when it comes to analysing each N-sized block, you will find that it's actually counterproductive; it will take up more time rather than saving some.
You can determine the correct key-length by computing the Hamming distance between the first key-length worth of bytes and the second key-length worth of bytes. Normalise the result by dividing by key-length. This is the first step to tackling this.
If you would like anymore tips or have any questions regarding how to proceed further, feel free to ask.