Try asking your childminder if she can recommend any colleagues who can offer cover over christmas? Is this a one-off decision for this year or will she always close at christmas from now on? Make sure she knows the situation with your work, she may have made this decision based on the assumption that everyone else wants time off at xmas as well.
If you're having trouble finding a registered person, you can legally use an unregistered person if your child is over 8 or if the childcare is provided in the child's home. Although you may not want to take the risk of using an unregistered person, consider if you have a close friend/trusted neighbour who could help out as a one-off? You could try a nanny agency for a temporary contract or sometimes parents have found help in local college students (if you find someone who is studying early years/education they may have already done a placement which required CRB checks) or preschool staff, classroom assistants etc.
Its quite common for childminders to close over christmas but you usually get a lot more notice!
www.emergencychildcare.co.uk
www.childcarelink.gov.uk
Your family may live 70 miles away but surely you'll be seeing them over christmas at some point? Could you tie in your family visits with a few days of them caring for your child in your home or theirs (carer doesn't have to be registered with Ofsted if related to the child). Eg if your family visited you for a couple of days then took your child back to their house for a week and then you visited them for a few days and then brought child back with you????
Good luck, hope you get it sorted out.