College student flights: Vacation in London first, then semester in Austria

She can do a multi-flight option with 3 legs - to London, London to Vienna, Vienna home.
She can do an Open jaw two leg flight, to London, and then Vienna-Home, and use any of the numerous Budget Air Carriers for a separet one way trip from London to Vienna. The Budget carriers usually make no distinction for a one-way trip, a round-trip is essentially two one-ways, no discount.

The main hassle I see, since I am not playing with the flights available to you, is finding a combo where her leg to London can be on the same flight as yours. You'll need to play with the matrixes on this.

Just carefully check fare rules. For a round trip (even with open jaw) the cheaper fares may have a max stay rule. For a student for a full semester, the time over there may exceed that..

We did this this past year with my son in St. Petersburg, no issue with flight there end of January and back end of May, and done as a pseudo 3-city so that he could stay in London (he was connecting there anyway) a few days on the way home. Only time issue was his Russian visa, as it's initially only 90 days, everyone in the program had to have them extended after they arrived. I would imagine there will be something similar for the 90/180 Schengen stay.

The main hassle I see, since I am not playing with the flights available to you, is finding a combo where her leg to London can be on the same flight as yours.

If I am reading this correctly, you all want to go to London, then you will return to the US from there and your daughter will continue on to Vienna. I don't know why booking all three of you on the same flight to London would be a problem. You would just book yourselves a round trip to/from London, and book her a multi-city arriving in London and leaving from Vienna to home. Then, as a previous poster said, just book a cheap flight from London to Vienna for her.

For budget flights in Europe, look at www.skyscanner.com and www.whichbudget.com The no-frills lines are built around single-flight sales rather than return trips. As another poster indicated, she should read the airline website very carefully and keep in mind that the opening price will inscrease through extra charges during the booking process. However, you get what you pay for and prices can be very low if booked well in advance.
Once those options have been considered, your daughter should check the trans-Atlantic airline to see if a cheap add-on between London and Vienna is available. Europe is so competitive that you never know what might turn up. Certainly she will be able to book Vienna-Detroit as part of her ticket to London (I'm betting Delta and its KLM/Air France partners.)
Be prepared for stiff prices across the Atlantic this summer which are currently at nose-bleed heights.

I agree that you flying round trip to London and the daughter doing a multicity stop or open jaw would be the way to go. Open jaw may be much cheaper, leaving her to find a way from London to Austria. Just beware of baggage costs on the budget airlines, I assume that since she is staying a semester she will have more than a 22 pound carryon. If luggage is a concern, look at Easyjet or possibly a conventional carrier.