database - How can this SQL query code be broken/exploited by user input?

We have a legacy app that doesn't do queries using positional parameters, and there's SQL everywhere. It was decided (before I started here) that since user input can contain apostrophes, every string input should be manually escaped for those apostrophes.

Here is the essential original code (not written by me), translated into C# for easier consumption:

private string _Escape(string input)

{
    return input.Replace("'", "''");
}

private bool _IsValidLogin(string userName, string password)
{
    string sql =
        string.Format
        (
            @"SELECT COUNT(*) FROM UserAccounts

                WHERE UserName = '{0}' AND Password = '{1}'",
            _Escape(userName),
            _Escape(password)
        );
    // ...
}

This really seems like it can be broken in some way, but I'm at a loss as to how it could be exploited by user input. Assume user input is unfiltered until it hits _IsValidLogin, and forget that passwords appear to be stored in plain text.

The solution to shore it up for good is obvious -- use positional parameters -- but I need some ammunition to demonstrate to management why/how this code is insecure so time/$ can be allocated for it to get fixed.

Note: I'm assuming this can be broken, but that may not actually be the case. I'm not a SQL superstar.

Note 2: I've expressed this question as database-agnostic, but if you can exploit this code for a certain engine, I welcome your contribution.