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.