This is not usually enough information to reconstruct the original plaintext message, unless advanced statistical techniques enter the picture.

Fortunately, you don't have to reconstruct the entire plaintext message from the two chunks of xored text. About 2N characters, not necessarily contiguous, is enough to start attacking the two original xor keys. It's pretty easy (and kind of fun) to do that much by eye. Just write a little perl script to plug in some common words, and see if what comes out looks like English text. Sometimes you'll guess wrong, but once you get three or four words that form a sensible phrase, things start looking pretty certain.

The standard technique for discovering an unknown key length is to try XORing the text against itself shifted at various distances, until you see a spike in the index of coincidence. The lowest common multiple of the spiked distances is then your key length.

Your method is probably better than mine. Told you I was an amateur.