Como dizer "Nem sei o que dizer" em inglês

A opção I don't know what to say é comum.

Paulo, you might want to take a second look at this one:

"They were so so lovely and kind towards everyone. [1] I don't have words to express how sorry I am!"

It's not quite ideal. You have no reason to say you're sorry If someone acts lovely and kind, that's not exactly very natural. The context here is that someone acts that way and then you say you don't know how to repay them for it or that you're so thankful that you can't even put it into words.

One way to express that:

B. They were so so lovely and kind towards everyone. [1] I don't have words to express how sorry I am!

[1] you could also just say "I am so sorry!".

You're right, Leonardo. I read it in haste and somehow I mixed the two events, A and B there. As if one had learned about someone's death (A) and the news was about someone lovely and kind towards him.

So, you are right. My bad!
Here goes my second stab at it then:

A. Ela morreu? Credo! Nem sei o que dizer. Morreu de quê? De Covid?
She is dead/she passed away? Oh my God! I am shocked! What she died of?
She is dead/she passed away? Oh my God! I am speechless! Sorry. How was that?
She is dead/she passed away? Oh no! I am shocked (and saddened) to hear that she died. Tell me how it happened?

The answer to your the second part of the question, proper, could be "do you know what she died of?"
But with the feeling of disbelief, and shock beyond words someone is likely to ask directly "it was from COVID?" or ask you to tell him/her how it happened, etc.

B. Eles foram tão gentis. Eu nem sei o que dizer.

B. Eles foram tão gentis. Eu nem sei o que dizer.
They were so kind, I can't express how thankful I am.

In a certain context, of course. I am not privy to the circumstances the sentence landed into the EE, though. "eu não sei o que dizer" can be something good or bad.

One instance of a 'umcomfortable' situation:
They could have been so kind, that I felt bad to give a two star rating to their inn/lodge. (I got myself embarrassed - "sem graça').

I think it is close to our "eu nem sei o que dizer/o que falar" (agora, neste momento). And it seems like having a stronger emotional and a "surprise" unexpected component to it.

Ref. phrasemix

But, mine isn't the last word, I would like to know what natives and advanced learners have to say on this one.

Thank you Donay for your insight. High five!

How about "I don't even know what to say or think" or "I don't know what to say".
They were so kind, I don't even know what to say.