You can reach out via email to ask for their best long term rate. A brand like Element or Marriott Executive Apartments will be more ready to offer a beneficial long term rate.
That said, depending on your daughter's level of comfort with disruption, it may be worth first booking a few back to back week long stays at different hotels before committing to a long term rate. The experience of living in a hotel long term can be fantastic if it's the right hotel. As a full time resident of hotels, I always have a short stay at a potential hotel first and only once I can see myself living there do I extend my stay. From bad luck with a room, to maintenance issues, to location, to vibe, there are all manner of reasons why a hotel might not be a good fit. A long term commitment to a less-than-perfect apartment is tolerable because finding an apartment is hard and time consuming and no apartment is perfect, whereas hotels are flexible by design so it can feel really annoying to be in a hotel that isn't right.
Another option is to book month-to-month, via the website, keeping one booking in advance. You'll typically find that the best "long term" rates on the website are for bookings of either 10 or 14 nights. That's the strategy I currently employ: I book 14 nights at a time, and when I check in, I immediately book my next 14 night stay. A stress free experience, while being very flexible. A long term commitment agreed with the hotel may come with restrictions (like not being able to terminate early) whereas when you're booking a standard flexible member rate, you can cut your stay short at any time. Hotel staff like people who stick around and are happy to minimise disruption by ensuring you keep the same room between stays (as it's less work for housekeeping).
From my experience, Element is more on the hotel side (a hotel room with a kitchen) whereas Marriott Executive Apartments are more on the apartment side (an apartment that happens to have some hotel-like amenities). I like Element, although I haven't stayed at either of their Doha locations. Unfortunately, though, the Marriott hotels (with apartments) are all ~10km from Qatar University so it's possible they might be too far away. I'm not sure how often your daughter would need to attend the campus, or whether she wants to be near campus for social reasons, so they may be ruled out on that basis.
Wow, this is great! Certainly a few things to think about and for her to consider. I appreciate the info. I was originally thinking just book it up until Winter break and then after the break (which should be 4-6 weeks) if she wanted, she could switch hotels. Her only other option is campus housing, that comes with a 10:00 o'clock curfew and a host of restrictions. She is willing to do that but I was thinking of an alternative and/or in case campus housing proves to be far too challenging.