Advice for 2 Night Getaway from Copenhagen

Rather than Malmö, you might consider Lund. It is a university town. Here is a post with some info from a couple of years ago.
https://community.ricksteves.com/travel-forum/sweden/lund-and-malmo

I think Helsingør and Frederiksborg (Hillerød) are best as separate day trips from Copenhagen. I did that and it was easy and I had enough time for dinner back in Copenhagen.

I don’t have much to help/offer with 2 night away.

So many choices!! I love Malmö, but it has many similarities to Copenhagen. And since it is so close, it will be easy for your daughter to explore on her own.

Other great train options include Lund or Ystad, Sweden. Both are lovely towns. I have not been to Odense, but many people love staying there for a few days.

I highly recommend Helsingør, plus the Louisiana Museum. However, as mentioned above, this is an easy day trip from Copenhagen vs. an overnight destination. But you could combine this with staying in the nearby Hornbæk/Dronningmølle area to experience the Danish seaside towns.

With a car, you could explore areas that are not too far away, but not as easily enjoyed on public transport. Going south, you could drive to Stevns Klint to see the church and lighthouse, and then onto the island of Møn, where there is a lot to do for a couple of days.

Going north, you could stop at Frederiksborg Castle, and then explore the seaside from around Tisvilde to Hundested. There are nice small towns, beaches, and landscapes in this area. From Hundested, there is a quick ferry to Rørvig, where there are cute shops and restaurants.

Indeed there are many good options. I concur with the recommendations given above.

I agree with the point that Helsingør (and Hillerød) and Malmö (and/or Lund) are so easy to reach from Copenhagen that they are great day trip options, and you can also get home late if you wish to spend the evening there before returning. In my perspective, the ease of getting there makes it not worth the hassle of moving hotels.

Maybe Aarhus is something for you? It’s a lovely city, yet modestly sized. There’s something for everyone there: History, culture, architecture, etc. I’m personally a big fan of Moesgaard Museum, and Den Gamle By is a fun experience as well. It’s about 3 hours from Copenhagen by train.

Odense, as mentioned above, can also be a good option - especially if you are interested in H. C. Andersen. Personally I prefer Aarhus, but there’s nothing wrong with Odense. It’s about 1.5 hours from Copenhagen by train.

Tickets to Odense and Aarhus are released 2 months ahead, so you’ll be able to book for mid-June in a few weeks. Tickets are usually reasonably priced if you book at least a couple of weeks ahead via dsb.dk or DSB’s app.

Møns Klint is also worth a visit if you want to get properly out of the cities, but as mentioned you should have a car for that.

Or for something a bit different, you could take the ferry to Oslo:

• Friday evening (16:30) to Sunday morning (10:00), Saturday in Oslo or
• Saturday evening (16:30) to Monday morning (10:00), Sunday in Oslo

https://www.gonordiccruiseline.com/minicruise/oslo-cruise/

You keep the same cabin onboard, so you can leave your luggage, but it is cleaned and refreshed like a hotel room. Price for a couple

• from 72EUR (Standard inside cabin with bunk beds)
• to 910EUR (Commodore De Luxe: Limited sea view, double bed, sofa bed, jacuzzi, wifi, breakfast)

Note

1. that the cheapest option is preselected, so if you want an upgrade you need to deselect that and select the one you want.
2. Premier/Commodore cabins are popular and sell out early.
3. Meals must be booked with the trip.

I'll make a second plug for Aarhus. Loads to see there including an excellent contemporary art museum. If you got a car, you could drive all the way to the northern tip of Denmark to Skagen and Grenen, where the two seas meet.
https://northabroad.com/grenen-skagen-denmark-travel-guide/
Lots to see along the way, as well. In mid-June, your days will be very long.