I take it the ceiling in the garage is plastered ?, it should be double tacked for fire and building Regs. I would say it's easier to take this down rather than pull up all the carpet/flooring in your daughters room ?.
Depending on what room is between the floor boards and garage ceiling will dictate what can be fitted in the space. You should be able to 100mm celotex in there or 150mm wool insulation, both will make it warmer.
The ceiling will need to be double tacked again to bring back to Reg's.
I've seen on some of the DIY programs in the US the spray on foam, that look like fun to do
. Not sure what that would cost to get someone todo ?.
I think makeing a false ceiling below the original will still give you a cold area between the floor and the new part ?