For now, the only button for Word to HTML conversion is the one you referenced in your message. That button was not developed by us and is part of the tinymce default icons. However, Josh and I have been analyzing the possibility of adding a similar feature with increased power and flexibility, but it's not going to be available any time soon.
In the mean time, this will get you out of a jam, with more than acceptable results:
1. Save your document as a web page (HTML) in Word ("filtered" or "normal" is your call, but do know that unfiltered HTML will include word-generated tags that will likely not validate properly);
2. If your documents have any images, Word will create a subfolder with all the resources from the document inside the folder the doc is being converted in, so upload all the resources you are planning on keeping to your uploads folder (through the media library of FTP).
3. Open your html inside Dreamweaver or any other code or text editing tool such as Notepad, Notepad, etc.
4. Delete all references to deleted resources in the generated HTML and update the paths to the ones you decided to keep. If you have PHP capabilities in your posts and pages, you can use get_bloginfo() so as not to use absolute paths.
5. Paste the HTML into the editor, in HTML (Text) mode, and save the post.
That should get you way closer than the "Paste from Word" button.