Technically, Harley Quinn has no bearing on the episode’s plot. The same story could’ve easily been told without her. What the episode does provide are just a few moments for her personality to come through. This is why Arleen Sorkin is so important. The character in her voice conveys both Harley’s playfulness and sense of danger. It’s all fun and games to Harley, regardless of who gets hurt.
She dons her villainous bodysuit, a chauffeur’s jacket, or a tight policewoman’s outfit, whatever’s necessary for the joke. When captured by Batman, she makes a possibly sincere crack about returning to beauty school. (Her origin hadn’t been developed yet, so who knows.)
Harley's affection for the Joker is conveyed only through the haircut she’s giving him in her debut scene. The audience has no way of knowing what’s next for the character. Chiefly, she’s there to break the monotony of large, beefy henchmen. Maybe there’s depth here, maybe she’s just a cool design. A humble entrance for a new character in a fledgling series.
And, following her modest debut, anyone with only a fleeting knowledge of popular culture knows what comes next. Harley Quinn is entrenched in the Batman mythos, ranking after only a few characters in public recognition. Now, what happened when Warner Brothers used this canon to introduce another new foe?