The Sentence's Logical Demand
The sentence has a single continuous argument running through both blanks:
The critic devoted herself with unflagging [blank i] to every manuscript regardless of its provenance or promise → this produced both virtues and vices → because the methodology was so [blank ii] as to border on the promiscuous

Blank (i) must name what she applied unflaggingly to everything. Blank (ii) must name what that indiscriminate application makes her methodology. The two blanks are not independent — blank (ii) is the consequence of blank (i) applied without discrimination.

Eliminating Each Wrong Answer
C. perfunctoriness - eliminated first and fastest. The sentence says her devotion was unflagging - meaning it never flagged, never diminished. Perfunctoriness means doing something with minimal care and effort, which is the precise opposite of unflagging commitment. A critic who reads perfunctorily is not devoted at all. This fails on the very first word of the blank's context.
B. assiduousness - the most dangerous trap for blank (i). It means diligent, persistent, careful attention - which sounds almost identical to zeal. The critical distinction is what each word implies about selectivity. Assiduousness implies methodical care - an assiduous critic pays careful attention, which suggests she weighs and distinguishes manuscripts. But the sentence says she attended to every manuscript regardless of provenance or promise - meaning she made no distinctions at all. Zeal is enthusiasm that can be applied blindly to everything; assiduousness is diligence that by its nature implies discriminating care. Only zeal leads logically to blank (ii)'s "undiscriminating."

D. polemical - means argumentative, confrontational, taking strong positions. A polemical methodology is actually highly selective - it picks targets and argues against them with force and specificity. But the sentence requires the opposite of selectivity: a methodology that treats everything equally, without distinction. Furthermore "a methodology so polemical as to border on the promiscuous" makes no semantic sense promiscuousness in its academic meaning refers to indiscriminate mixing, not to controversy.
E. desultory - means lacking consistency, jumping randomly from thing to thing without sustained engagement. This is tempting because desultory methodology sounds like it could apply to everything randomly. But the sentence has already established in blank (i) that the critic's engagement was unflagging - sustained and persistent. Desultory directly contradicts unflagging: you cannot be both consistently devoted and randomly inconsistent simultaneously. The two words cancel each other out.

Why A + F is Exact and Complete
Zeal means intense, enthusiastic, sustained commitment. The sentence says this zeal was unflagging (never diminishing) and applied to every manuscript regardless of provenance or promise (without selection). Zeal applied indiscriminately to everything is the precise character trait the sentence is describing.

Undiscriminating means failing to make distinctions between things of different quality or merit. When you apply unflagging zeal to every manuscript regardless of promise, you are by definition not discriminating between good and bad manuscripts - you are treating them all the same. The methodology is therefore undiscriminating.

The phrase "as to border on the promiscuous" seals it. In its academic, non-sexual sense, promiscuous means "without discrimination, indiscriminately mixed." This is the exact synonym of undiscriminating - the sentence is essentially saying the methodology is so undiscriminating that it approaches promiscuousness. No other option in column (ii) shares this meaning: polemical means confrontational, desultory means inconsistent. Only undiscriminating fits.