DON'T get into that complicated conversation with her, OP!
She suckered you. By playing quiet and withdrawn, then you took the bait and asked the question that hooked you and reeled you in with the crying performance.
Then when the opening £3k didn't land, she pivoted to her next goal of £1k. Testing to find your trigger sum. What she might be able to get out of you.
If you do get into the conversation as you described, she'll play you again by seizing on any mention of the word 'gift'.
This is how con artists operate. By observing and testing, looking for weak points (ie your compassion and kindness), then apply pressure and twist.
Shut her down.
Just say 'no'. That is all you need to say. 'No'. No apologies, no explanations, no anything she could seize on.
It's not your problem, not your duty or your job to offer to fix it, or to suggest solutions for her financial issues, if they are real.
Direct her to speak to the line manager, HR, staff welfare. People whose job it actually is to support staff wellbeing.