There are two actual questions you could be asking:
- What should one say to a child whose pet has just died?
- How can I rectify an apparently-bungled approach to a discussion with a child whose pet has just died?
You have given information which suggests that your relationship with your daughter is not as strong as you'd desire: a cat you've never known (they have lifetimes generally well over 10 years), you do not know (and apparently cannot ask) her current address, and the only means of communication is Facebook (but if she can access Facebook, there is a high probability she has phone service and you cannot access her by that means).
I do not point this out to be mean to you -- far from it. I point it out to set the context of her view of things.
Death itself is not formulaic, nor is coping with death. The answer to #1 above is, "it depends." The other answers have given good guidance, but there is not a single answer to the question.
Now, for the rest of this answer, I'm going to assume you understood what she was saying in the broken dialog (I wouldn't have initially caught her intention.)
When someone effectively says (for whatever reason), "My cared-about-thing died and I don't want to talk about it", saying, "I am sorry for your loss" is cliché in this day and age: it shows no compassion and effectively says, "I don't know what to say, so I'll apply a rubber stamp response to your dramatic situation."
So, then, how do we say the truth without making the situation worse: that we do not have the words we want, but care so very much? The answer is the truth:
Oh, no! I know there aren't the right words, but when you want to talk, I'll listen.
That 1) validates their emotions, 2) tells the truth, and 3) opens a gateway of communication for when they feel the time is right. Yes, I just gave a single answer to what I said could not have a single answer, but therein lies the "trick"... by saying truly no more than, "I care; talk when you're ready", you provide all the different answers! As has been said by others, the greatest conversationalist is the one who is the best listener.
The assertions that you were focusing on yourself in other answers are erroneous. You were expressing yourself in the best way you knew:
- you said that you were sorry for her loss (response #1);
- you said that she was a great parent to her pet (response #2);
- and you said that you were sorry that your comments caused her more grief (response #3)
-- at no point were you intending to talk about yourself!
Now, on to answering the second potential question. I wrote "apparently-bungled" on purpose. Emotional people are almost never logical and when the height of emotion passes they are better able to reflect. You have an advantage here: Facebook chats don't delete, so they aren't forgotten or mis-remembered.
The end of your chat, though, ended the chat, so now you will need to open the door again. This, too, is not a one-size-fits-all answer, so here are some suggestions:
- Happy New Year! How are things over there?
- Hope you're doing well! Hit me up when you've got time -- I'd love to hear what's new in your life.
- Just dropping a line to say that I love you.
The primary idea: talk about something other than the last topic of conversation -- she'll bring that up if and when she wants.
Finally, there is the chance of a "backfire" where she asks, "How could you have been so insensitive about 'Fluffy'?" The answer is always the truth:
I am sorry. I wanted to have the right words and tried to show how much I care, but it didn't go the way I hoped. How are you now? I'm here to listen.
And, then, of course, do as you said: listen -- it's the best thing you'll ever say.