Ten Broeck, who quit as a full-time professional golfer to become a full-time caddy for PGA Tour member Tim Herron 4 years ago, shot a 2-under 68 at the Indianwood Golf & Country Club for a 36-hole total of 134.
Kite, 62, started on the back nine and shot a 4-under 31, but suffered four bogeys the rest of the way. His 70 left him alone in second place at 135 in his bid to become the oldest winner in the history of the event.
There were 24 players under par for two rounds.
Ten Broeck said he felt his caddying experience was a benefit to him while competing in a major championship.
"I think it helps me with my strategy or just overall attitude on the golf course as far as trying to press," Ten Broeck said. "You know you have to play safe sometimes."
Defending champion Olin Browne survived the cut at 3-over 143.