Cannot find correct connection string

Well, no - it won't work!
Firstly, LocalDB is a specific name which means "on this computer" - and unless you have installed SQL Server on both computers (which I don't recommend) the connection string must refer to the specific computer running the instance of SQL server holding the database you need to connect to.
Secondly, you only attach databases for development - you cannot attach a DB in production as "full" SQL server doesn't support it. You should add the DB to SQL properly, and connect to it, never try to attach except when you are developing the software.
Thirdly, Integrated security may well be a problem - unless you have specifically permitted your domain to access the "target" computer, it won't work. And it's a bad idea anyway, you should create users with only the relevant permissions: INSERT row, UPDATE row, DELETE row. Integrated security gives full permissions, which means that they can edit or delete tables in your DB.

We can't tell you exactly what to use: we don't have access to your LAN.
But these are the two strings I use for my local access:

Connection=Data Source=GRIFF-DESKTOP\SQLEXPRESS;Initial Catalog={0};Integrated Security=True

And my tablet:

Connection=Data Source=GRIFF-DESKTOP\SQLEXPRESS;Initial Catalog={0};Persist Security Info=True;User ID=MyUserName;Password=MyUserPassword

Where {0} is replaced with the DB name.