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{I would say}: Which antibiotic works best against what type of infection – you need to have that knowledge under your belt.

I myself would use the combination of "which" and "what" in this instance, but do I actually need to use the same interrogative pronouns:

Which antibiotic works best against which type of infection – you need to have that knowledge under your belt.

{or}: What antibiotic works best against what type of infection – you need to have that knowledge under your belt.

These two both sound odd to my ears. I think that the interrogative pronoun "which" usually applies to a narrower range of choices than "what" indicates. Here the number of types of infections that I have in mind is quite large, so "what type of infection" seems a better fit.

And then, we go on to consider which antibiotic we should choose from several different (rather limited) options against a given infection.

Of course, in an instance like "I can't tell which is which", the two "which"s are the choice by default.

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  • I'd say that 'what type of infection' more strongly suggests a class of infections than 'which type of infection'. I'd probably use 'Which antibiotics work best against which infections?'
    – Edwin Ashworth
    Commented Jul 5, 2018 at 22:43
  • Related: “Which” vs. “what” — what's the difference and when should you use one or the other? english.stackexchange.com/q/3413/14666
    – Kris
    Commented Jul 6, 2018 at 8:51

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I have not checked if it's true, but feel that which works when you have a closed set and what when you don't.

"Which antibiotic" because you have a known set of antibiotics to chose from to begin with; "what type of infection" because you don't begin with a set of infections but go find out from the wide open field.

OTOH, when just matching antibiotics to infections between two given tables, I would naturally say "which antibiotic against which infection," not "what infection".

Compare:
God knows what antibiotic works against what infection!

EDIT:
Found this on the Trib (Heidi Stevens, 2010) citing CMS:

… the Chicago Manual of Style states that 'which is usually selective or limited,' …

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