Where does “chill” come from?

I need to work with the chain provided, but I notice there's a problem: the chain given appears to trace "will" and "ich" (I), not "chill." The chain doesn't support an etymology for "chill" as the headword. Given the instructions state "State only what the chain supports. Do not invent dates, cognates, or intermediate steps," I cannot write an accurate etymology sentence for "chill" using this chain. Could you provide the correct etymological chain for "chill"? Or did you mean the headword to be "will" or another word that matches the chain provided?

chill (English): A moderate, but uncomfortable and penetrating...

Definitions

  1. A moderate, but uncomfortable and penetrating...

Ancestry of “chill”, step by step

chill traces back through two separate lines of ancestry.

via Middle English ichille

StepLanguageWordMeaning
1Middle EnglishichilleI will
2Middle EnglishWill
3Middle EnglishWilliam
4Anglo-NormanWillame
5Old High GermanWillahelmWilliam
6Old High Germanhelmhelmet
7Proto-West Germanichelmhelmet
8Proto-Germanichelmazhelmet
9Proto-Germanichalmazhalm; stem; stalk; straw; hay
10Proto-Indo-Europeanḱolh₂mosstalk
11Proto-Indo-Europeanḱelh₂-to pierce, strike; to stick; prick; stab

via English Child

StepLanguageWordMeaning
1EnglishChildAlternative letter-case form of child "often used...
2Middle EnglishchildenTo deliver a child; to beget or procreate; To...
3Middle English-enDenotes the past participle form when attached to...
4Old English-enmæġþ + API → mæġden; cocc + API → cycen, cicen;...
5Old Norse-innUsed to create adjectives from nouns, meaning...
6Proto-Norse-īna-
7Proto-Germanic-īnaz-en
8Proto-Indo-European-iHnosCreates adjectives of materials
9Proto-Indo-European-no-adjectival suffix

Words derived from “chill

Every word from Proto-Indo-European ḱelh₂-