Didn't you pronounce い as a long vowel, like "ee" in "eel"? い in ない/すい is a short vowel, like "i" in "ill". Thus, ない is like "n + eye", as you wrote, but not "na + ee". See the audio file in the page linked below.
0:20
Betsudan fukai riyuu demo nai.
1:02
Hikaru koto wa hikaru ga, kiresou mo nai.
http://en.wikibooks.org/wiki/Japanes...#Normal_speech