Honestly, for your example, I'd just say "...until she collapsed too." "[On/to] the floor" is sort of implied, especially because I assume we already knew Chisato was on the floor.
But if you were going to specify the floor, either of your options can be correct. "To the floor" feels more like it's referring to a change in where Chisato's mom is - she was on the couch or standing something, and then collapsed to the floor. Or it feels like it's trying to emphasize how completely she collapsed: not just to the chair, or down to one knee; she collapsed all the way to the floor.
"On the floor" doesn't really care where she started collapsing, just that she is now on the floor (having collapsed there). It's giving information about where she is and how she got there, but less trying to emphasize how far she fell.
- "She collapsed on the floor": the information I'm getting is that she's now on the floor
- "She collapsed to the floor": the information I'm getting is that she fell all the way to the floor, not just partway
It's hard to describe because I think these are just shades of meaning--both are correct.