This is not nice: you send us not the code you are using, but something else. For example, this line would not compile:
SmtpClient client = new SmtpClient;
This is not that helping you is impossible. The problem is different: if we help, we should assume that you could be trusted. Nobody wants to waste time. Who knows what else in your code is different from the text you show in your post? And if we don't know, why wasting time on looking at your code?
There is another reason for not taking your question seriously. If you real code compiles (as you reported), it means it does exist. You did not bother copy and paste it properly. Have you been uncomfortable with that? If you get help, how can we be sure that you are able to use it?
I would not spend so much time on writing all that if I did not want to help. Of course, I am trying to help, but I want to play fairly. Let's respect each other's time and effort.
As to the problem: 1) there a a lot of code samples around; so you can always find the simplest code sample on the Web and compare your code with them; 2) how can you be sure that your SMPT server is installed properly and authentication is correct? you can check it up using any available working mail client program.
This should be quite enough for you to fix the problem. Please try.
—SA