With some generous assistance from someone I was able to do a board level repair which fixed this laptop. The RA77 resistor had disappeared from the board somehow, which resulted in a broken circuit,  and that was preventing the audio codec chip from being powered. I ordered a suitable replacement and soldered it on using a hot air station. Now the audio completely works.
I am working on a Dell XPS 15 9560, Win 10, which has no Realtek audio device. It is not listed in Device Manager. Intel HD Audio is there and I do get sound if I connect an HDMI cable to a TV but the laptop itself will produce no sound and I get no sound from the headphone jack. There is no audio device to select for default output and recording in the setting in Windows. I ordered a replacement audio board from eBay and installed it but still no sound. I pressed F12 to enter the diagnostics system and tried testing the sound there but there was no sound generated. I made sure that the laptop's internal speakers are plugged in. I checked the UEFI settings and made sure that the audio system is enabled. I tried disabling and enabling it just to be sure. All Windows updates and optional drivers have been installed and I tried installing the Realtek driver provided by Dell for this model of laptop but still no Realtek audio hardware is listed in Device Manager. What can I do to get the sound working again?
XPS 15 9560 with no sound