To add a listener, you must have a reference to a DOM Node - not a string. Here's an example:
// Get a element to set the innerHTML contents on
var div = document.createElement('div');
// Add html created by strings
div.innerHTML = '<span style="float: right; margin-right: 25px; margin-top: 4px;">'+
'<img id= "' + downloadLeft_ + '" style="position:absolute;" ' +
'src="/public/images/excel.gif"></span>';
// Add the div to the document
document.documentElement.appendChild(div);
// Get a reference to the image element
var img = document.getElementById(downloadLeft_);
// Add listener
goog.events.listen(goog.dom.getElement(downloadLeft_),
goog.events.EventType.CLICK, function() {
//event listener code
});