It appears that the authentication mode can be determined/changed by checking the value of a registry key.
Within Management Studio, if I change my local database from Windows only to Mixed and script the change to a new query window, it generates the following query.
If you do not have SSMS installed, you might still have sqlcmd installed. Invocation would be something like
sqlcmd.exe -S MyServer -d master -Q "EXEC xp_instance_regwrite N'HKEY_LOCAL_MACHINE', N'Software\Microsoft\MSSQLServer\MSSQLServer', N'LoginMode', REG_DWORD, 2"
You might need to use xp_regwrite vs xp_instance_regwrite depending on whether you have multiple instances on a server (my example does). This article on sswug shows it using xp_regwrite.
If you don't even have sqlcmd/osql etc installed, you can edit the registry directly but be certain to back it up before making changes.
Finally, the instance needs to be restarted before the change takes effect.