One of the reasons MySQLi is more "secure" is because it offers a different interface, which is better in many ways. Instead of trying to translate your code directly, learn the new interface and use it. If that's all your code, it wouldn't be easy to rewrite from scratch, and which is more important, look up the equivalents (and alternatives) for everything you're doing in the code that you pasted.
For starters, you should use $mysqli->prepare
with parameters instead of interpolating variables like you're doing.
http://www.php.net/manual/en/mysqli.prepare.php