Since the OP's goal is about once a week, that means she can choose fair-weather days for her commutes. That simplifies clothing selection quite a bit.

When I first starting riding a bike to work 20 years ago, it was 8 miles, not 13 but like the OP's my goal was once a week. I rode in my work clothes which included a tie. I rode on days that weren't too hot or too cold and with little chance of rain. I didn't carry anything with me. I only did it when it was going to be light both ways.

You could argue that a MTB with knobbies is not the best for that distance but my cruising speed is maybe 2 to 4 mph faster on a road bike vs an MTB. The difference in average speed is less than that. It's going to take her some extra time but not an order of magnitude difference.

She'll be fine.

Yeah, she might have a bad experience and decide that it's not going to work after a single attempt. She might make that same decision after spending another $200-300 on lights, tires, and clothes.

Her biggest challenge will be learning how to deal with the traffic on her commute and maybe conditioning. Twenty six miles in one day can be tiring if you're not used to it. But both of these will get better over time, regardless of clothing or equipment.