The letter writer doesn’t mention gastrointestinal symptoms so that rules out a bit. Lyme disease is not present in Peru (or most other countries in South America except for Brazil as far as I know) as the ticks are different – soft versus hard.
I’m assuming all the normal tests have been done to rule out things like malaria, as Richard mentioned, you could consider bringing up Mayaro virus or Western or Venezuelan Equine Encephalitis (don’t let the name fool you, both affect people too). It might be worth it to go to a travel clinic with an ID specialist where they may be more familiar with stranger diseases. The peculiar thing about your disease is the seemingly long incubation period, making it seem like Lyme, but that is highly unlikely. Have they tested you for Lyme? Is it possible you contracted it before or after your trip?