Other answers have already correctly identified that you are right and your friend is not.
I wanted to point out that you can interpret this as:
There was not a line in her countenance, not a note in her soft and
sleepy voice, but [the ones that] spoke of an entire contentment with her life.
Basically, "all the wrinkles on her face are from smiling, and every inflection in her voice is a happy one." Here 'but' just means 'except' (which is still pretty common). The archaic thing is omitting the relative pronoun. The pattern of omitting a relative pronoun to create a subjectless relative clause was quite common in Middle and Early Modern English, but is nowadays restricted to sentences with indefinite-pronoun antecedents, existential sentences, and cleft sentences. (See this reference which I've not read in full (it being paywalled), but which I'm including for the citation of Quirk et al. 1985, A Comprehensive Grammar of the English Language, regarding relative pronoun elision). Affirmative, rather than negated, examples include things like "There's a table stands in the corner" or "There's a pain goes on and on."
I emphasized 'existential sentences' above because the type of sentence you are citing here is an existential sentence. (I had written this up in the first draft of my answer but deleted it since I didn't want to get sidetracked... oh well.) In the example 'There was not a line... but spoke of contentment', part of the meaning of the sentence is to stress that some lines do exist--specifically, the ones that speak of her contentment.
User ruakh does well to point out that the omitted relative pronoun needs to match the stated antecedent in number, even if the sentence is logically describing multiple people. For instance, his example from Shakespeare's Comedy of Errors, "There's not a man I meet but doth salute me," while the speaker (Antipholus) presumably meets many people in a crowded market, replacing the omitted relative pronoun should give "There's not a man I meet but [he that] doth salute me." (This is in line with other uses of a singular to refer to multiple people individually, even though they are logically plural, e.g. 'England expects that every man will do his duty'.)
Thus, "There's no man but [he that] serves the king" and "There's no men but [those that] serve the king" would both be grammatical; "There's no men but [those that] serves the king" would be incorrect. Note also that the (somewhat fussy) rules of joint subjects apply: when the subject is joined with 'and', it is a joint subject, so "Every father and mother love their child" --> "There's no father and mother but love their child" (plural), but when 'or' or 'nor' is used as the conjunction, the overall subject is treated as having the number of the final antecedent. So:
- "The officer or infantrymen serve the king" is correct, thus "No officer nor infantrymen but serve the king"
- "The infantrymen or officer serves the king" is correct, thus "No infantrymen nor officer but serves the king"
- "The infantrymen or officer serve the king" and "No infantrymen nor officer but serve the king" are both incorrect.
As you can probably tell, this comes up so rarely that examples to demonstrate it are rather contrived.
User Accumulation is correct that this is somewhat archaic, but I think many native speakers would understand it if they noticed the word 'but'. (I mean I think it is unusual, but still productive in a linguistic sense.) You will see the same construction in phrases like "No god but God" or "No war but class war". The first example is the title of a book that was written in 2005, and the latter phrase is an anarchist slogan that still shows up in graffiti, so people clearly expect others to understand this grammar structure. If you do choose to use it, though, you might want to keep it brief, so listeners don't miss the 'but'. Note also that, being already the product of omitting some words, the structure lends itself to heavily elided phrases such as slogans and somewhat poetic statements.