If anyone remembers Stabler nearly left his wife for Olivia at one point as the 2 were heading towards a divorce. Kathy got pregnant and Stabler stayed with her due to that in 2000's. But Kathy always knew there was a chemistry the 2 had for each other. That they gave off a husband and wife vibe together due to the closeness they had. So she of course would make sense to write the letter due to her feelings on it.