Looking through the Koyo Gunkan, I found the section on the death of Takeda Shingen:
"On the 11th day of the 4th month around 1pm, Lord Shingen's condition took a turn for the worst. His pulse became extremely rapid. On the night of the 12th, approximately 9pm, he developed an abscess/rash in his mouth, and 5 or 6 of his teeth fell out. He gradually weakened."
There is no mention of him getting shot by a sniper, a somewhat popular theory, probably due to the movie "Kagemusha". It appears that he died late in the night of the 12th of the 4th month of 1573 (5/13 by the Western calendar). According to the Koyo Gunkan, in addition to giving his last will and testament, he requested that his death be hidden for three years, which he felt would discourage anyone from attacking Kai. He also advised Katsuyori to make peace with Uesugi Kenshin, believing Kenshin to be an honorable man who would not refuse.
There is mention of a previous sickness that afflicted Shingen prior to his death. I intend to look at that next.