You need to install to the doc root, which by default in the TKL LAMP appliance is /var/www. The landing page (aka Turnkey Control Panel) that you get with the TKL LAMP appliance is /var/www/index.php (webroot: myname.tklapp.com/index.php). So if you replace that with the root of the WP files then it should just work...
The other thing you could do is just start with the WP appliance and reconfigure that to be multisite...
I suggest start again...
You need to install to the doc root, which by default in the TKL LAMP appliance is /var/www. The landing page (aka Turnkey Control Panel) that you get with the TKL LAMP appliance is /var/www/index.php (webroot: myname.tklapp.com/index.php). So if you replace that with the root of the WP files then it should just work...
The other thing you could do is just start with the WP appliance and reconfigure that to be multisite...