Unfortunately, I don't have a single clue. These log files are so extremely verbose, and the same time so void of useful information. This is the passage where things go wrong:
MSI (s) (1C:10) [09:16:25:146]: Running as a service. MSI (s) (1C:10) [09:16:25:157]: Hello, I'm your 32bit Elevated Non-remapped custom action server. <Func Name='SNAC_SetClientSSNLDefaults'> Setup: Succeeded to do the netlib client setting CSsetDefaults(FALSE). <EndFunc Name='SNAC_SetClientSSNLDefaults' Return='0' GetLastError='0'> MSI (s) (1C:F0) [09:16:25:190]: Invoking remote custom action. DLL: C:\WINDOWS\Installer\MSI9DBB.tmp, Entrypoint: SNAC_SetClientSSNLDefaults MSI (s) (1C:C8) [09:16:25:190]: Generating random cookie. MSI (s) (1C:C8) [09:16:25:193]: Created Custom Action Server with PID 18068 (0x4694). MSI (s) (1C:B8) [09:16:25:216]: Running as a service. MSI (s) (1C:C4) [09:16:25:219]: Hello, I'm your 64bit Elevated Non-remapped custom action server. MSI (s) (1C:20) [09:16:25:226]: Note: 1: 1723 2: SNAC_SetClientSSNLDefaults_64 3: SNAC_SetClientSSNLDefaults 4: C:\WINDOWS\Installer\MSI9DBB.tmp CustomAction SNAC_SetClientSSNLDefaults_64 returned actual error code 1157 (note this may not be 100% accurate if translation happened inside sandbox) MSI (s) (1C:20) [09:16:25:229]: Product: Microsoft SQL Server 2012 Native Client -- Error 1723. There is a problem with this Windows Installer package. A DLL required for this installation to complete could not be run. Contact your support personnel or package vendor. Action SNAC_SetClientSSNLDefaults_64, entry: SNAC_SetClientSSNLDefaults, library: C:\WINDOWS\Installer\MSI9DBB.tmp MSI (s) (1C:20) [09:16:25:229]: Note: 1: 2318 2:
Error 1723. There is a problem with this Windows Installer package. A DLL required for this installation to complete could not be run. Contact your support personnel or package vendor. Action SNAC_SetClientSSNLDefaults_64, entry: SNAC_SetClientSSNLDefaults, library: C:\WINDOWS\Installer\MSI9DBB.tmp
NET HELPMSG 1157 returns One of the library files needed to run this application cannot be found. I don't if the error number in question is relevant with NET HELPMSG, but at least it matches the verdict A DLL required for this installation to complete could not be run. But which file? No idea.
One possibility is have Process Monitor running when running the installation to figure out which that may be missing. But it is very tedious, and the error could be in execution of msiexec.
It may be better to raise the white flag with regards you computer itself, and install create a virtual machine and install SQL Server on the VM. If you have never worked with virtual machines before, there is certainly a hurdle. But knowing how to work with virtual machines is an asset in itself, so it is well-spent time. Not the least than compared to chasing ghosts with Process Monitor.