You said that your code does not work. But you did not say how the code fails.
Also, you have not shown real code. You have shown only a few lines.
I can try to help, but you need to (1) Show more code. and (2) Describe the exact failure. If it is an exception, then show the exact exception.
This is what it would look like in C#, to read a single entry.
using (var zip = ZipFile.Read(pathOfExistingZip))
ZipEntry e = zip["\\path.to\\file\\in\\archive.txt"];
Stream s = e.OpenReader();
// read stream here ...
I don't know exactly how to say that in VB.NET, but you should be able to figure it out.