I don’t think you should even be thinking in terms of a sensible ‘contribution’ towards her fuel. You need, at very least, to cover her full costs. I assume that you would expect her to have business use included in her insurance before allowing your child in the car with her. It costs a lot to properly maintain a safe vehicle, and even the 45p per mile still agreed by HMRC is on the low side.
She has collected and dropped off your child presumably 5 days, so 10 return journeys? That’s 10 times safely strapping your child into a car seat, and getting him out, etc.
£120 is £12 per return journey, so £6 for a single journey. It might be a 5 minute drive, but it will clearly take her longer than that, walking to her car, securing your child, walking him to your house at the other end, etc.
I am sure you hoped for a lesser amount, but it was fairly unrealistic to think that a professional child minder would provide her time for free. With the wonderful benefit of hindsight it would have been better to have discussed the cost beforehand, but lesson learned for the future.
Would you have been able to get anyone else to take your child to and from the childminder for a lower cost if you had known what she intended to charge?