The title is pretty self-explanatory. I was listening to the new History of Rome Appendices from Mike Duncan and he mentioned that it's not that hard for one person to read literally everything we have from ancient literary sources from Rome, and that's kind of sad. But it also made me think about how many primary sources we must have for a lot of other things. So anyway, the Battle of Noryang is something I'm interested in knowing more about, and I wanted to find out if there are primary sources translated in English I could access.
This is somewhat of a late response but since you asked for primary sources I don't believe you got a good answer. They're not about the battle specifically but provide a contemporary perspective:
Jingbirok ("The book of corrections"). This was written by Ryu Seong Ryung, who was prime minister prior and during the war and key to Yi Sun Shin's war efforts. He wrote that after the war as an analysis of what went wrong and how it could've been prevented. Generally speaking I regard this as a must for anybody with serious interest in the war.
Nanjung Ilgi ("War diary of Yi Sun Shin"). Obviously he died in the battle of Noryang so he didn't write about it but this diary provides strong context of the war up to that point, how he went about things, and what he was as a person.
EDIT: Forgot to mention "Imjin Changch'o: Admiral Yi Sun-sin's Memorials to Court". These are the formal reports made to the royal court by Yi and compliment his diary.
A quick side comment on Samurai Invasion and The Imjin War would be that both Turnbull's work and Hawley's work are derivatives that apply their own interpretation. Turnbull's work is mostly military focused while Hawley goes for a "narrative" approach. Unsurprisingly they have significant issues with each other's work.
Personally I've read both and found Turnbull's work to be more objective than Hawley's because he takes in both korean and japanese sources while Hawley does come across as pushing a specific agenda in his story telling (e.g. painting people in a certain way, overly emotive approach, etc) and I have difficulties taking it seriously as a history work. Like many other works about korea in english, I'd read it as a chronicle of events but avoid taking the opinions of the author as gospel.
EDIT: Further to my comments about author bias, see the review of Hawley's work and Kenneth Swope's "A Dragon’s Head and a Serpent’s Tail: Ming China and the First Great East Asian War, 1592-1598"
The two books authored by Hawley and Swope, respectively, contain countless factual errors, fictionalized accounts, and outright fabrications, all stemming from the quality of the sources upon which they relied. Almost every page of these two books contains some form of mis-information. The authors fail to rely upon firm empirical data on the Imjin War. Having failed in this most basic and fundamental task, their analyses could not help but be fatally flawed. It is as though they decided to cri-tique Shakespeare’s Richard III (which is believed to have been written in 1592), without actually reading it, by choosing, instead, to rely on second-hand comments, anecdotes, and gossip, to which they cavalierly add their own guesswork.
from pg 20 (just prior to Epilogue) of Nam-lin Hur's "Works in English on the Imjin War and the challenge of research", International Journal of Korean History (Vol.18 No.2, Aug. 2013)
Unfortunately agendas and narratives are distressingly common in english works about korea.
Check out Stephen Turbull's Samurai Invasion and Samuel Hawley's The Imjin War. Those are the two most common books in English on the Imjin War and their list of sources would be the best place to start. Unfortunately, Hawley was born and raised in Korea so a lot of his sources are undoubtedly in Korean and Turnbull's specialty is Japan so his sources are probably in Japanese.
Good luck!