DatePicker on Visualforce in HTTPS not using standard style

Question

I've inherited an externally facing form which has a date field on it that pulls up a date picker when selected. We are converting to HTTPS and since the whole page is totally custom, the standard SF calendar popup doesn't work and they coded one in. Unfortunately using HTTPS breaks it and though the date field still works, I can't get a datePicker calendar to pop up. I've tried using just the standard VF inputField and it doesn't work. Been scouring any article I can find and not having any luck finding this issue being addressed. One mentioned that it may be the API version, but jumping back to an old API version didn't make a difference.

Any ideas on how I might be able to get this working without needing to re-write the entire page would be highly appreciated.

Here is the datepicker function (init runs on page load)

function initDatepicker() {
    var vdate = $$( "input[id$='date-of-visit']" )
    if(vdate.length){
        vdate.datepicker({
            changeMonth: true,
            changeYear: true
        });
        vdate.attr('readonly', true);
        $$( "input[id$='date-of-visit']" ).click(function(){
            $$( "input[id$='date-of-visit']" ).datepicker('show');
        });
    }
}

And where it is being used in the VF form

<div class='input-container clearfix date-of-visit-container'>
    <apex:outputLabel for="date-of-visit"> Date of Visit
        <apex:outputText rendered="{!pkbCon.isFeedBack}" >
            <font color="red" >*</font>
        </apex:outputText>
    </apex:outputLabel>

    <div class="customInput date-of-visit">
        <div class="left"></div>
        <div class="middle">
            <div class="top"></div>
            <div>
                <apex:inputField id="date-of-visit" showdatepicker="false" value="{!pkbCon.theCase.Visit_Date_and_Time__c}" onchange="validateDate(false)" />
            </div>
            <div class="bottom"></div>
        </div>
        <div class="right"></div>
    </div>
</div>

Answer 1

Thanks to @Eric for the suggestion of locating a possible js library that was loading in http rather than https, turns out there was one that was buried and I had missed. Fixing that reference allowed the datepicker calendar to popup when the date was clicked.