Would need a better look at the hens comb, side shot of body to better tell what breed the mother is. From the top view of her comb it looks pea. The far shot of her body and head looks like an easter egger. What color were the eggs?
You must be fairly close to your neighbor as one hen wouldn't lay so many eggs to be viable hatching. I mean for 14 to hatch that would be a large clutch of what? 20+ eggs? Meaning if hers would be 20+ day old eggs there. I think others aided her in making the clutch and she went broody. I'm guessing she is not the sole mother.
You should offer your neighbor something for the hen. Say she's been hanging out on your property and you've become attached to her. The chicks wont need anything with her there but I feel the mother is your neighbors. As in if a neighbors dog (you know who's dog it is) started hanging out on your property you'd not fence it in and call it yours. If they want the hen back just supply a light to the chicks for heat until they feather out. Or offer some chicks to them for renting their hen to raise your chicks for 3 weeks then give her and some chicks back...dunno, some ethical solution can be arranged. And who knows? They may not care about their hen having chalked it up to predator loss when in reality she went broody on a hidden clutch.
Edit to add: I'm also thinking if she and her chicks were not penned in now she would proudly trot them back to introduce to her flock. Honestly I'd ask the owners their thoughts and express your attachment, stewardship and see what comes of it.
