What verb can I use to describe someone who is talking very quickly?

How about:

sputter

To speak quickly or in a confused way because you are upset, surprised, etc. (Merriam Webster)

Or Gabble

To talk quickly and in a way that is difficult to understand. (Merriam Webster)

Her eyes widened. "Jonathan," she cried, "what happened to you?"

The reporting verb, to cry suggests that the speaker spoke instantly and showed deep concern at the same time. If the OP prefers to emphasise the "rush" and speed of her speech, then I would suggest babble

(Oxford Dictionaries)

babble: Utter something rapidly and incoherently

‘Thank goodness you’re all right,’ she babbled

If the OP wants to convey shock, anxiety, and fear; the noun alarm could be used

Her eyes widened in alarm. "Jonathan," she cried, "what happened to you?

Other reporting verbs can be found on CDO and Wiktionary

I'd like to convey how surprised she is to see him again and how desperate she is to help him.

So instead of:

Her eyes widened. In a rush, she stumbled out, "Jonathan, what happened to you?"

Consider using:

Her eyes widened. Without hesitating, she eagerly inquired, "Jonathan, what happened to you?"