You can add a login in this case with DAC.
DAC is by default disabled - use below instructions to enable it.
1) Launch Services Applet in Control Panel
2) Stop SQL Server Express Service
3) Launch Properties dialog of SQL Server Express Service
4) Add "-T7806" to the startup parameters
5) Start the SQL Server Express service
Then use -
SQLCMD -Usa -Ppassword -S.\SQLEXPRESS -A
or in Management studio - (Over server name put below and place SQL login etc.)
ADMIN:.\SQLEXPRESS
- Add the login you want to add now.
But, my doubt is -
By default - BUILTIN\Administrators logins found in each instance - (unless you drop this login). So, local administrators are always having rights to connect the instance.
Check event-viewer of machine and SQL Server error-log to find the root cause. There might be something for you troubleshoot.