I have a web page in Bulgarian and I want my users be able to translate it one-click to English. Also there should not be any translation banner at the top of the page when
Google has thought ahead my friend. Please look at this page: http://translate.google.com/translate_tools
EDIT: I'm sorry, I just realized you're using what the page provides! You can, with simple javascript, hide the elements that are displayed and create a link for English where it's onClick changes the value of the hidden select element...and your entire page is translated.
It's a bit messy but it gets the job done and the user doesn't know it exists!
You can also consider capturing the request that is sent to the google translate servers and capture the link that is called when you select English and just use that link.
Chrome has a nice utility to capture requests (see ctrl+shift+j for a developer console)