It will be my pleasure to cite the author of the letter. I'm looking at perhaps ten sentences out of a four page letter.
I have the actual letter, so the quotes will be accurate.
The letter is from one doctor to another; I am the subject.
I got the letter by snooping through my medical records. The author of the letter probably has no idea that I have seen it. As a matter of fact, I doubt that she remembers me. The letter is ten years old. It's a small part of a book-in-progress.
The answer will depend on the laws of your country.
In general, the rights of the sender for privacy etc. have to be weighed against the public's need to know.
Private communication is private and in most cases must not be published, because otherwise the personality rights of the sender are being violated. But if that letter contains information that would help solve or prevent a crime, the need of the public to know of that crime outweighs the rights of the sender.
Similarly, commercial letters are to be kept confidential if they contain information that the sender has a right to keep confidential, but if the letter contains information that is of importance to the public (e.g. unlawful business practices), that information might outweigh the right for confidentiality.
Commonly, only a court of law can decide if publishing (parts of) a letter was lawful.
If you want to avoid breaking the law, you can anonymize the information in the letter, so that there is no possibility for anyone to identify the sender (and in this case, receiver) of the letter. That is, even your family who know of your visit to that doctor must be unable to conclude that it was this or that doctor who wrote (and who read and reacted or failed to react appropriately) to the letter! If you cannot completely anonymize the quote, and the quote would throw an unfavourable light on any of the persons involved, I would consult a laywer before publication.
If the quote is harmless and would not be considered slander by anyone, even the doctors themselves, an anonymized publication seems unproblematic to me.
This is certainly a question best answered by a lawyer (ideally one with knowledge of the area) However...
The critical question you need an answer to is who owns your medical records. Medical record ownership varies from state to state, according to healthinfolaw.org1 Alaska (guessed from your name!) has no laws governing ownership of medical records
If you are the owner of your medical records, then you should also own the copyright, and be perfectly entitled to publish any part of them. If a medical institute owns the records you would likely need permission before you would be legally entitled to publish. (although it would be difficult to imagine a court upholding a case against someone publishing their own medical record - but don't quote me on that)
The real answer though, talk to a lawyer before you do anything, understand the risks before you pursue a course of action.
(It may be worth asking that question on the law site of stackexchange)
