SQL Server mixed mode authentication not being set
To start, even though the gui shows mixed mode authentication, running

SELECT CASE SERVERPROPERTY('IsIntegratedSecurityOnly')
WHEN 1 THEN 'Windows Authentication'
WHEN 0 THEN 'Windows and SQL Server Authentication'
END as [Authentication Mode]


gives

'Windows Authentication'


Two related small things resulted in this issue. (1) My windows domain account is not an administrator, so I can't restart the SQL Server instance from the Configuration Manager. (2) Restarting windows does not reload the SQL Server configuration options, or at least, it won't change the authentication setting.  

A windows domain account with administrative privileges on my local machine was able to stop and restart SQL Server from the Configuration Manager. I can now login with the SQL account I created.   

After that, the above query gives  

'Windows and SQL Server Authentication'


So, even though I can change the authentication options, and SQL Server displays in the GUI that the options have been changed, the options are not actually changed until the SQL Server instance is manually restarted by an admin (rebooting doesn't do it).