Accessing UserControl's DataContext from another UserControl
Based on what I understood from your question here are my 5ct.  

The MainPage class is the parent class of the LoginClass.  


Parents should talk to children by using methods.  
Children should talk to parents by using events.  


So you should raise an event from the LoginClass to be consumed by the MainPage instead of accessing a property of the MainPage.  

In this way you can easily test the code of the LoginForm independently of the MainPage whereas in you current implementation a MainPage always needs to be created before the LoginOperation_Completed() method can be used.  

In addition in its current state your implementation has side effects because each time you create a new MainPage the Instance property will be changed.