Just thought of something: if I understood correctly, Reynir can see ghosts, but not "healthy" spirits. This may mean that if Tuuri is infected, she may reach a point where her spirit becomes visible to Reynir, possibly before the physical symptoms start showing up. Because of policy of keeping them apart, he may notice only once it's well underway.
As for magical resolution, I have an idea of what it may be on the Icelandic magic front: you remember how the anti-ghost rune was originally a rune keeping sheep from straying too far away from the farm? Reynir may be able to figure out the "keeping the spirit inside the body when it really really wants to leave" variant, with the catch that the person has to wear the thing 24/7. Perfect for a temporary fix that will need to be improved upon by people who actually know magic. Shut up about the potency of the cure after ninety years, little chemistry education that I have! I guess that's what Lalli takes care of, or something?