How can I connect to a SQL Mirror Database?

This may not be a real solution to your issue, but WHY do you want to connect a Notebook to a Mirrored SQL database? Typically, you use Notebooks to collect and process non-tabular data sources (Excel, JSON, XML, etc.) and from there dump it into tabular destinations (read: tables). 

If data is ALREADY in tables in the mirrored database, why not use T-SQL to access and process it?

because you can do things more easily and in a saner way in Python than with the antedeluvian 4GL language that SQL is.  That would be my reason #1.  

#2 you got a bunch of libs in Python to do data science and data engineering that do not exist in the SQL world.  

and on and on it goes.

I have solved this issue by creating a User data function. This process is explained here in Microsoft Documentation.

Python programming model for Fabric User data functions (Preview) - Microsoft Fabric | Microsoft Learn

This allows me to make the connection and fetch query results. I am able to get views as well as tables.
As of now, I haven't figured out how to do the stored procedure execution and that is next in line.

my function looks like this -->

import pandas as pd
import fabric.functions as fn

udf = fn.UserDataFunctions()

# Select 'Manage connections' and add a connection to a Fabric SQL Database 
# Replace the alias "<alias for sql database>" with your connection alias.
@udf.connection(argName="sqlDB",alias="MYMIRRORDATABASE")
@udf.function()
def read_from_sql_db(sqlDB: fn.FabricSqlConnection, query: str)-> pd.DataFrame:
    '''
    Description: Read data from Mirror DATABASE using input query.
    
    Args:
        sqlDB (fn.FabricSqlConnection): Fabric SQL database connection.
    
    Returns:
        DataFrame query result.

    '''

    # Establish a connection to the SQL database
    connection = sqlDB.connect()

    df = pd.read_sql(query, connection)
    print(df.head())
    return df;