I agree with Astralbee's mention that everyone grieves differently. There is no surefire way to correctly approach every grieving person.
It may have been inevitable that she was going to respond the way she did to your question. That's no one's fault, though I can see why you'd construe thaty ou caused her outburst.
However, one tip though:
I asked a simple "how are you doing?". It proved too much - she broke down and sobbed for quite some time.
Think about the answer to "how are you doing?". The correct response to the question is describing her current state of mind, which requires her to think about how she's dealing with her father's passing, which requires her to think about her father's passing.
While the question comes from a good place, trying to answer it does the opposite of what you wanted it to. You wanted to distract her from what's been bothering her; but your question explicitly pulls it into focus.
Given the situation, it would've been better if you'd approached her on a topic completely unrelated to her father or her current state.
The fact that she was attending a birthday party proves that she at least tried to find a diversion, even though she wasn't able to actually focus on the party.
I'll use the example of my grandfather. He lost his wife (my grandmother) two years ago and was inconsolable fo over 18 months. Any attempt to console him would only cause him to cry harder.
My aunt focused on telling him how much we all love and miss her. That just made him cry more. My mom focused on where to go from here (he wasn't able to take care of himself, partly due to age partly due to the grief). That just upset him even more. My cousins brought his greatgrandchildren over, but since my grandmother was usually the one to play with them, they kids reminded him of her.
So I took a different approach, and talked to him about looking for a new place to live. He had been a landlord (as a side job) for a few decades, and has always been an instinctive advice giver.
He talked about tenant laws, explained some things to look out for, and started telling me stories of horrible tenants he had in the past.
He was perfectly capable of telling a funny story and laughing at it (not as loud as usual, but still smiling), because he was doing something familiar (giving advice and talking about what he knows) that did not in any way touch on my grandmother.
The same applies to the neighbour girl. Ask for her advice (which puts her in the role of having a handle on things) with something she's very knowledgeable on (so she can feel like she has a handle on things), e.g. if she's a good cook, ask her if she knows how to improve an average recipe of yours.
It's about empowering her, so she doesn't feel like she's a slave to the circumstances (which is often the root cause of grief) but rather in control of the situation.
More often than not, once you get them talking, they revert to their old selves. for as long as the current topic is their main focus.
If she does find her way back to the sad parts ("my dad always liked my cooking..."), try to acknowledge what she says but reframing the core of her message ("From what I hear, everyone likes your food. My attempts at cooking, however, ...").
People often have different mindsets that they're able to switch between. Some people are different at work than at home. Many people I know, when visiting a childhood friend, often revert to behaving the way they did back when they were close friends.
You can subtly (without her realizing) making her shift her mindset.
She likely won't switch her mindset willingly (it can be interpreted as dishonoring the person who died), but if you can do it without her realizing, it is likely to give her a reprieve from her grief, which is what she most needs. You can't fix the death of her father, but you can offer her a short moment to focus on other things in life.