If you are using asp:DropDownList its possible that jQuery is not finding you element by it's ID("ddl_Division") due to asp.net webforms always messing with controls ID, to resolve this you could recover the element ClientID
var ddl_Division = <%= ddl_Division.ClientID %>
and in your script
$("#" + ddl_Division ).append("<option value='" + Division_ID + "'>" + Division + "</option>");
or you could add a class "dynamicPopulated" for an example to your dropdown and doing the append this way:
$(".dynamicPopulated").append("<option value='" + Division_ID + "'>" + Division + "</option>");