I honestly don't think they analyse the private conversation as much as you'd think. I sent a hundred or so pages of private chat and according to the notes, they shredded a big chunk of it due to "excess chat logs." I'd focus more on evidence like joint mortgage/lease, bank accounts, and other evidence showing you live at the same address. As for evidence of a genuine relationship, you can submit photos, letters, greeting cards, affidavits from family, friends, and associates, fb posts, etc. And yes, go ahead and write a letter. Do you have any other evidence of contact like snail mail letters or cards? You must have something somewhere.