As a follow up:

After tinkering around I was able to connect to my home network, but I didn't do anything different.  Then I noticed this odd behavior:

1. The Wi-Fi connection would only show up in APCC-FIND if I unplugged the LAN cable and refreshed FIND

2. If I initialized using "serial" in APCC I could not connect to the CP5 Wi-Fi using SkySafari Pro

3. I could connect to the CP5 Wi-Fi but only if I initialized using SkySafari Pro (iPhone) or the LAN connection, but the SkySafari connection was very sketchy. Any slew commands resulted in a runaway scope.  I had to do an emergency stop using APCC.  SkySafari locked up in the button pushed position and even doing a hard close didn't stop the slew.  Timeout was set for 3 seconds in SkySafari.

4. Even when the settings page showed I was connected to my home network, I could still not connect or initialize the mount via Wi-Fi using APCC (2000ms timeout set).  The Wi-Fi address would not show up using APCC-FIND, and APCC would not connect when I manually entered the IP shown on the settings page as connected to my home network.

Subsequent power cycles resulted in the Wi-Fi module not connecting to my home network as described in the original post. When I would try to manually connect through the setting page I would often get an error which says, " 169.254.0.102 says Join Failed, General Error."

This Wi-Fi module is definitely flakey.  It seems like it isn't initializing upon power-up most of the time, and when it does initialize it still doesn't operate correctly.