Twig combined conditionals to set variable
It's probably better if you set variables to reference, so your if statements make more sense.

{% set asset = story.storyImage.first() %}
{% set videoUrl = story.videoStoryUrl %}
{% set icon = "text" %}

{% if asset | length %}
    {% if videoUrl is empty %}
        {% set icon = "video" %}
     {% else %}
        {% set icon = "image" %}
    {% endif %}
{% endif %}

{{ icon }}