My code that I need to end up with is
myOutput += "<img src ='~/ImageHandler.ashx?id='" + ID + class='person-image'></img>"
You can add double quotes using the escape character
myOutput += "<img src=\"~/ImageHandler.ashx?id=" + ID + "\" class=\"person-image\"></img>"
This will append
<img src="~/ImageHandler.ashx?id=35" class="person-image"></img>