TL;DR: from the name Khitan, a Mongolian tribe which once ruled Northern China; got to Russian from Mandarin via a Turkic language.
Китай: From Old East Slavic Китай (Kitaй), from a Turkic source; compare Kazakh Қытай (Qıtay), Kyrgyz Кытай (Kıtay), Uzbek Xitoy, Turkmen Hytaý; from Old Turkic Qïtaǰ “Khitan”, from Mandarin 契丹 (Qìdān, “Khitan”), the Chinese name of a Mongolian tribe who ruled Manchuria from 907–1115 CE.
Cognate with Spanish Catay and English Cathay.
I put down ,"Have you ever visited China" and got it wrong. Is there something I missed?
I wouldn't expect it to work, to be honest, while the meaning is the same, you're using words that are totally different.