I don't like dialogue chunks, so I tend to do what you did . I have a few sentences of dialogue that tapers off into out-of-dialogue narration, or interruptions from characters, if it's natural for them to pipe in with questions (if someone is telling me a story, for example, I tend to listen to the whole thing first and don't seek clarification until later. Others need clarification right away, and it helps break up paragraphs. It can also feel more natural).
Summary
"When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals."
The goats were her favorite, she explained, and Bobby, a lone pygmy goat, was the best of them all. Her father would say he never quite acted the same when she wasn't around. [story continues]
The second option is to have dialogue blocks. When you decide the paragraph of dialogue is too large, you start a new line, but don't end the paragraph with a quotation mark. For example:
Summary
"When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals. When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals.When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals.When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals.When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals.
"When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals.When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals.When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals.When I was a little girl, my father would take me to the family ranch every summer. I remember being so excited because I'd waited all year long to see those animals." <--- This is the end of the character's dialogue, so now the ending quotation mark goes here.
The fact that your story is in third-person shouldn't affect the first-person nature of Katrina telling her entire story in dialogue. There's already precedent for it as every single character throughout anyone and everyone's novels speak in first-person when talking about themselves unless they're royalty or something. Personally, it'd be as normal to me as a comma.
If this is a central character, who's one of the frequent POVs, a third option that I see around is to start a new chapter with the story as a trip down memory lane. There's multiple ways to do this depending on style, but in short, it's like this:
Summary
Chapter 5 ends with Katrina beginning her story. One can taper off with ellipses if that's their style, or give some other transition indicating she's still telling her story.
Chapter 6 begins from the perspective of Katrina as a little girl (you indicated this is third-person, so as a reminder, it may be a good idea to keep the tone the same and not reminiscent of a little girl as the narrator of your novel is its own 'character' that does not change in tone). Preferably, it's clear within the first paragraph that she's younger now, so anyone can piece together right away this is a flashback to tell the story.
Find something that fits your style and feels most natural to you :). I like any and all ways depending on what style works with the story. You can even use multiple style for different instances of storytelling from other characters. Good luck!