Checking your linked site, you may include a script tag passing a ?var=desiredVarName parameter which will be set as a global variable containing the IP address:
<script type="text/javascript" src="http://l2.io/ip.js?var=myip"></script>
<!-- ^^^^ -->
<script>alert(myip);</script>
I believe I don't have to say that this can be easily spoofed (through either use of proxies or spoofed request headers), but it is worth noting in any case.
In case your page is served using the https protocol, most browsers will block content in the same page served using the http protocol (that includes scripts and images), so the options are rather limited. If you have < 5k hits/day, the Smart IP API can be used. For instance:
<script>
var myip;
function ip_callback(o) {
myip = o.host;
}
</script>
<script src="https://smart-ip.net/geoip-json?callback=ip_callback"></script>
<script>alert(myip);</script>
Edit: Apparently, this https service's certificate has expired so the user would have to add an exception manually. Open its API directly to check the certificate state: https://smart-ip.net/geoip-json
The most resilient and simple way, in case you have back-end server logic, would be to simply output the requester's IP inside a <script> tag, this way you don't need to rely on external resources. For example:
PHP:
<script>var myip = '<?php echo $_SERVER['REMOTE_ADDR']; ?>';</script>
There's also a more sturdy PHP solution (accounting for headers that are sometimes set by proxies) in this related answer.
C#:
<script>var myip = '<%= Request.UserHostAddress %>';</script>