Check the error log which usually under the folder C:\Users<youraccount>\AppData\Local\Microsoft\Microsoft SQL Server Local DB\Instances\v11.0%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
.