... he was too sick to tell whether or not I was his son, ... (Para. 13)
Meaning: ...he was so sick that he could not tell whether or not I was his son.
This use of the structure "too ... to" is shown by the following examples:
It's too cold to go swimming. 天气太冷, 不能去游泳。
It's too good an opportunity for you to miss. 机会太好了, 你千万不能错过。
He's too much of a coward to fight. 他太懦弱, 不敢抗争。