Doesn't this sentence say "their daughters" instead of "their daughter"?

Yes, it does.
We should omit the "lar", and change the k to the corresponding "yumuşak g" (ğ).

hmm now I'm not so sure... the ending for "their" is -leri/ları"...

http://www.turkishlanguage.co.uk/possadj.htm

is "aile" plural or singular?

Oh, that's interesting.
aile ("family") - maybe it's considered plural in Turkish. I'm not sure about that (I'll have to look for example sentences with "aile" in my Turkish dictionaries (or on the Web).

I inquired a little...

Jones ailesi kızlarını seviyor.
The Jones family loves their daughter.

But, if I'm not mistaken, it can also be construed as:
The Jones family loves their daughters.

What do you think?
We need your help, Turkish speakers.

Jones ailesi kızını seviyor.

This is another sentence which I received from a friend of mine. I believe now there is no ambiguity.

The only thing that remains is to receive a confirmation from one of our colleagues here.

You are very fast.
Jones family loves their daughter.
Jones ailesi kızlarını seviyor.

As you know there are hidden subjects in Turkish:
My daughter: kızım
Your daughter: kızın
His dauhter: kızı
Our daughter: kızımız
Your daughter: kızınız
Their daughter : kızları

İf you say "Jones ailesi kızını seviyor",it means "Jones family loves your daughter."
kızımı-kızını-kızını-kızımızı-kızınızı-kızlarını.
Collective nouns are both singular and plural in English, such as family, team. The same in Turkish. Here, "aile" is singular as a whole. But possesive adjective should be plural. Take care.

I think the explanation above is enough for both of you, right?
Hasta la vista.

Yes, thank you! :)

@Duran
Bu çok ilginçti. Teşekkürler!

@Duran,
Thank you very much, from me too.
Bu gerçekten çok ilginçti.

By the way, after reading your comments, I wondered about the following:
kızları - could that also mean: his/her daughters?
And if so, "their daughter" and "his/her daughters" at the same time?
Türkçe zor bir dildir ☺

@Eldad
("Türkçe zor bir dildir.") Doğru!!

Tamy,
Evet... ;-)