What are the best entry level integrated amps and speakers?

By all means don't purchase an amp and speakers without auditioning them in person, if that's possible. I'm sure you would advise your daughter about that. Some amps and some speakers match up well and others do not.
Here's a first hand example....
I used to use a HK receiver with a pair of Klipsch monitors. Those speakers weren't the lower of the Klipsch line but their top Reference series. The sound was OK but I decided to upgrade. I purchased a NAD integrated (C375BEE) and connected the Klipsch speakers. Sounded like a thick blanket had been thrown over the speakers and sounded much worse than with the HK receiver.
Then went to my local audio store and auditioned my NAD amp with a pair of PSB Imagine B speakers. The improvement of sound was astounding. So, proper amp/speaker match ups are essential.

+4 NAD/PSB setup. If you can get a used NAD375BEE with used PSB Imagine T speakers (that might cost a wee bit more than $1500 but not much more), your daughter will be in high cotton. Floor standers that match well with the 375. My sons use this setup in their room. I confess that I go up there and listen when they are spinning the vinyl because it sounds pretty darn good.

So much of it is personal preference.  My bedroom system is a pair of B&W CDM2 Signatures I got on eBay for $350 and an old Arcam integrated amp I have had for 10 years or so.  I threw on some bulk Transparent Audio Labs wire I got a long time ago.  I guess they sold it for rear channels?  For $500 total system value, it sounds awesome.

Emotiva BasX preamp and Amplifier.  150wpc both for $600.  No bass management, but leaves enough to get very good speakers that don't need a sub.

Elac is getting so much press for being top performing gear, you must pay that some credence. Perhaps 50% of your budget goes to the speakers.
Right now on audiogon there is a steal on a classe integrated amplifier with a small defect. Original price $2000+, selling for six and change.

There is also an amplifier available now on Audiogon from Peachtree. It is their nova integrated with built-in DAC. I had a peach tree deco for a while, and it was great. The nova is even better! It is selling for 600 something.

add a pair of entry level kimber Or audioquest speaker cables for $150 and you have a fantastic system for $1500!