Admittedly, I noticed
Someone may say this 確かに modifies 違っている, but I think it's easier to think this 確かに as a sentence adverb. The main topic of the whole sentence is 私 (i.e., 妻's husband), which is omitted.
??? is a fairly long noun sentence, where 看護師 is the topic.
Admittedly, I noticed [that the nurse was different in her way of working].
"AはBが違う" literally means "As for A, B is different" or more naturally, "A is different in B".
Admittedly, I noticed [that the nurse was somewhat different from others in her way of working].
は in 他の人とは is contrastive (and thus optional). 一味 is a nuanced word meaning "small but significant", "somehow", etc.
Admittedly, I noticed that the nurse whom my wife wanted to look after [me/you/our son/etc] was somewhat different from others in her way of working.
妻が「ついていてほしい」と言った adjectivally (not adverbially) modifies 看護師 as a relative clause. (lit. "the nurse about which my wife said 'I want her to be with [you/me/our son/etc]' ")
So-called thematic は can appear in a noun clause like this (e.g, 彼は学生だと思う = I think he is a student), but not in a relative clause which modifies another noun (e.g. [×]彼は食べたケーキ [○]彼が食べたケーキ). That's why が is used after 妻 and は is used after 看護師.