You may not enter the US on a non-immigrant visa if your intent is to stay here permanently. It sounds like she told them she was going to get married. She could have been barred from entry. You can either marry her in Denmark and apply for a green card or your can bring her here on a fiancee visa and marry her here. Base on your timetable the second alternative is probably the best.
I agree with the other lawyer's comment. Of course, she may obtain another student visa if she has bona fide educational pursuits (since u said she is writing a thesis). Then she would be here on a legitimate non-immigrant ground, whereupon you could also get married. Always seek the advice of a lawyer before applying for any immigration relief.
You could be prevented from adjustment based on "immigrant intent" when a non-immigrant, student or tourist, marries while present based on a non-immigrant visa. You may be approved, but you do take that risk. A key issue if the adjudicator reviews it, is if the immigrant had immigrant intent before the entry.
If you marry in Denmark, and then attempt to enter the U.S. as a married couple, your new spouse may be prevented from entering the U.S. -- unless you filed for an immediate relative visa package through the U.S. Embassy in Denmark, and were issued an immigrant visa (Green card, conditional, 2 years subject to lifting the conditions based on marriage to a USC). Yes, this makes it very difficult for couples trying to have dual weddings!
Yes, she was lucky to be allowed in-- but the reason was probably that her intent was not viewed as immigrant
intent at the time of the entry. Each non-immigrant visa can have limitations of time and purpose. A tourist visa, unlike the student visa, is typically 6 months, and not duration of status. A tourist visa should be marked as DUAL INTENT if there is also a student aspect involved!
You could file for a fiance' visa for your fiance'. Then, if approved, the Missouri Service Center sends an approval notice, and your fiance' files paperwork and obtains a visa to enter the U.S. and you marry within 90 days and file for adjustment (An I-485 package, including I-765 and I-131 advance parole for travel) documents.
You don't want to LEAVE the US while in Adjustment of Status in certain visa categories or you can lose the status and have the petition abandoned. Therefore, if you did want to have dual weddings, you might have the second wedding during the adjustment period, after you have the advance parole document in your hands.
As far as a quick turn around time, and an "emergency," a wedding, even if booked, paid for, and planned, unfortunately is NOT an "emergency" to most adjudicators of the I-131 form. As life varies, it would depend on the adjudicator reviewing the petition. The short version, get approvals in your hands first, finalize the plans for the non-U.S. wedding second... or visa process in Denmark after marrying there before trying to re=enter and marry in the U.S. It seems like a lot of process to have to go through for your joyous occasion, and it can be--
but it's good practice, because marriage is all about working together as a team through both good and bad times.