Cannot start SqlLocalDB instance with my Windows account
Check the error log which usually under the folder %localappdata%\Microsoft\Microsoft SQL Server Local DB\Instances\v11.0 will give you big hints.

Log back in as the administrator account (not an administrator), and start a command prompt with Run As Administrator. Then run this:

sqllocaldb share v11.0 MyInstance


You might want to explicitly add your Windows account as a sysadmin on this instance if that is a goal. So:

sqllocaldb start MyInstance


Then connect with SQLCMD, Management Studio, what have you, and run:

CREATE LOGIN [Domain\Username] FROM WINDOWS;
EXEC sp_addsrvrolemember N'Domain\Username', N'sysadmin';


Now log back in as your Windows account, and you should be able to start this instance using:

sqllocaldb start MyInstance


(It may already be started depending on your O/S and how you switched accounts.)

Then Domain\Username should be able to connect using SQLCMD, SSMS etc. using (localdb)\.\MyInstance.