How did you find her, OP?

First I would decide when you need her to babysit. THEN, I would choose 2 OTHER days for a block or 2.5 hours each and give her a comprehensive list of "musts" for those days with "additionals" at the bottom if she has time. I would try taking 2.5 hours to do the list yourself to be sure it can be done and that you are being realistic. That way, she's not ignoring your kid to get her other work done so that she can work fewer hours overall. Make it clear to her that while she is babysitting is not the time to be cleaning the house.

It's not "rent free" if she's working for it.

I would personally just give her a daily/weekly housecleaning list of things you want done. Let her no you want it done when she's NOT babysitting.

Obviously her interning comes first then you

So I would set times that work with her schedule.
Our housekeeper comes twice a week 10-3. So you could say Tuesdays 9-12 & Fridays 2-4 do this this and this. I wouldn't over structure it like 9-930 sweep. Also don't expect her to just sweep and dust on any old day just because it's dirty or you're to tired to do it.

Plan the housekeeping like having a housecleaner coming to your home on a set schedule and a babysitter who comes to your house on a set schedule.

Number of hours and specific tasks. Then optional tasks, time permitting.

We are doing a room exchange with a girl who will be interning in DC- she will be living with us rent free in a nice space in exchange for 15 hours of babysitting and housekeeping per week. It looks the breakdown will probably be about 10 hours babysitting and 5 hours housekeeping. Is there a particular way that I should structure the housekeeping? I essentially want the 5 Hrs going towards stain treating, washing, drying, folding and hanging up my daughter's laundry, keeping the playroom tidy and organized, and maybe sweeping the ground floor floors a few times a week if there is time. Just not sure how to structure this? Do I set specific hours for it or just give her the jobs?