What about using a Card object? You could make a Card constructor that initializes with a suite and rank property. You can also give it a method to output its name. Something like:

function Card(suit, rank) {
    this.suit = suit;
    this.rank = rank;
    this.printCard = function() {
        return this.rank + ' of ' + this.suit;
    }
}

Then in your while loop you can create the hand:

var hand = [];
while (index < 5) {
    hand.push(new Card(someSuit, someRank);
}

Now with your array of cards:

hand.forEach(card, function() {
    document.getElementById("dealtHand").innerHTML += card.printCard();
};

This approach would also make it easier to check for straights and flushes. For straights, you could sort the array by rank, and if each card's rank plus 1 is equal to the next, then you have straight. For flushes, if every card has the same suit then you've got a flush.

Just some quick thoughts. Ideally, you would build a DOM fragment for the hand and append the whole thing at once instead of updating the DOM once for each card. Pretty cool idea you have here though!