I was trying to find the documentation for this but it seems to have been moved. I am assuming you have deployed the ASE environment with the internal load balancer option here?
One way you can access this is to log into the Azure portal from a virtual machine hosted on the same virtual network as your ASE environment, this should allow you to access the Kudu console as you are accessing it internally from your virtual network. You may have to modify your hosts file to point the SCM URL to the internal load balancer private IP address of your ASE.
Here's another possible method but I have not tried this myself: https://www.chenlong.hu/blog/read?id=VqmoyN&category=Tech