How did Kirito find out Zekken's secret? What did Kirito whisper during his fight with Zekken before which he was  defeated? In episode 20, Asuna asked Zekken (Yuuki) about this and she replied that Kirito had "figured out [her] secret".

What was the secret?
In episode 19, Asuna reports that Leafa thought that Kirito had had an exchange with Zekken during their fight. Kirito then tells Asuna about what happened:

Kirito: Um, something was bothering me.
Asuna: What?
Kirito: I asked, &quot;You're totally living in this world, aren't you?&quot; Zekken answered with a wordless smile and a charge skill at incredibly speed. That speed was beyond the limits.
Asuna: &quot;Totally living in this world&quot;? Do you mean they're a really heavy user?
Kirito: No. I wasn't referring to a single VRMMO. I felt like Zekken was a product of the full-dive environment itself.
Asuna: What does that mean?
Kirito: I don't want to give you any preconceived notions. I want you to feel the rest for yourself.

(Text from Crunchyroll subtitles.) Thus, that she is &quot;totally living in this world&quot; must be the &quot;secret&quot; that Yuuki is referring to; we do not hear of Kirito having any major interactions with Yuuki at any point prior to their duel. Furthermore, Kirito's intuition is correct. Spoilers for the rest of the arc:

 In episode 22, Kirito states his guess that Yuuki is a Medicuboid user and directs Asuna to the only hospital in Japan using the device.
 We then learn that Yuuki is an AIDS patient. She has been using the Medicuboid machine experimentally to alleviate the suffering caused by her health problems. Beyond providing pain relief and alleviating sensory impairments, the Medicuboid can also immerse the user in a virtual reality environment, such as that of the video games being played in the SAO universe.