How do I use reflection to invoke a private method?

Simply change your code to use the overloaded version of GetMethod that accepts BindingFlags:

MethodInfo dynMethod = this.GetType().GetMethod("Draw_" + itemType, 
    BindingFlags.NonPublic | BindingFlags.Instance);
dynMethod.Invoke(this, new object[] { methodParams });

Here’s the BindingFlags enumeration documentation.