Per Derksen, Vasmer and Chernykh, from Proto-Indo-European*kʷōy-, from the root *kʷey-. Cognate with *cěnà(“price, value”), which is an exact reflex of Ancient Greekποινή(poinḗ, “penance, penalty”), Avestan𐬐𐬀𐬉𐬥𐬁(kaēnā, “penance, penalty, punishment, revenge”). Also cognate with Sanskritचयते(cáyate, “to avenge, to punish”), Avestan𐬐𐬁𐬌𐬌-(kāii-, “to repent”), 𐬗𐬌𐬐𐬀𐬌𐬌𐬀𐬝(cikaiiat̰, “to cause to repent, to avenge, to punish”), Ancient Greekτίνω(tínō, “to cry, to repent”) (Homeric τῑ́νω(tī́nō)).
Per Rix (LIV), cognate with *čàjati(“to expect, to wait”), from a different Proto-Indo-European root *kʷey-(“to perceive, to observe”) (this root is reconstructed as *kʷeh₁y- by Derksen).
Chernykh, P. Ja. (1993) “ка́яться”, in Историко-этимологический словарь русского языка [Historical-Etymological Dictionary of the Russian Language] (in Russian), 3rd edition, volume 1 (а – пантомима), Moscow: Russian Lang., →ISBN, page 390
Trubachyov, Oleg, editor (1983), “*kajati (sę)”, in Этимологический словарь славянских языков [Etymological dictionary of Slavic languages] (in Russian), numbers 9 (*jьz – *klenьje), Moscow: Nauka, page 115
Vasmer, Max (1964–1973) “ка́ять”, in Oleg Trubachyov, transl., Этимологический словарь русского языка [Etymological Dictionary of the Russian Language] (in Russian), Moscow: Progress