You would be better off turning your HTML into a form, with its action attribute pointing to http://mysite.com/events.aspx and its method attribute as get.
Then you should rename your keyword input box as kwd. Then you wouldn't need to use any JavaScript to get it to work.
However, if that's not possible for some reason...
jQuery
var keywordInput = $('input[name="keyword"]');
$('input[name="SearchButton"]').click(function() {
window.location = 'http://mysite.com/events.aspx?kwd=' + encodeURIComponent(keywordInput.val());
});
Without jQuery
var keywordInput = document.getElementsByTagName('keyword')[0];
document.getElementsByTagName('SearchButton')[0].addEventListener('click', function() {
window.location = 'http://mysite.com/events.aspx?kwd=' + encodeURIComponent(keywordInput.value);
}, false);
Remember that < IE9 doesn't support addEventListener(). You'll need to use attachEvent().