There is LAPI.html in the same folder. It has some example which I am also trying to use but so far only managed to connect to a session preconfigured in my Liaison installation.
Here is my code so far, which I simply created in an .hta file.
<script language="JavaScript">
    var objLAPI = new ActiveXObject("LiaisonAPI.ctlLAPI");
    if (objLAPI)
    {
        document.write("<div>LAPI Created</div>");
    }
    else
    {
        document.write("<div>LAPI Failed</div>");
    }
    var sessions = objLAPI.GetSessionNames();
    objLAPI.Session = "DYNQAA";
    var objConn = objLAPI.Connect();
    if (!objConn) 
    {
        document.write("Connection Failed");
    } else {
        document.write("<div>Connected</div>");
    }
    var strDynPID = objLAPI.DynamicPID;
    document.write(sessions + strDynPID);
    document.write("<hr>");
    objLAPI.Disconnect();
</script>
It work, but I am yet to figure out how to post actual commands to the terminal and read out its output.
If you do succeed, please kindly let us know.
Thank you
Regards
Alex