Unfortunately, that quick posting bookmarklet can only be used for publishing on Tumblr.com - there's no way to integrate Tumblr with your existing WordPress or Blogger Blog.
However, here's some good news for WordPress bloggers - there's a new Tumblr inspired bookmarklet for WordPress that allows you to quickly post photos, videos and blocks of text from web pages to your WordPress blog just like Tumblr. See Screenshot:
You install the Tumblr Quick Post plugin like any other WordPress plugin and it will provide you a bookmarklet that you can add to the browser toolbar.
When you click the Quickpost bookmarklet from any web site, a new window will pop up. The URL of the page you're on, linked with the title of the page, will automatically be transferred to the editor in that window. You can do almost everything you do with the default editor of WordPress and create a new post for your blog.
The only issue is that if you tumblelog a photo on the web, this Quick Post plugin will hotlink to that image - that's different behavior from Tumblr where a copy of the image is uploaded to your Tumblr account.