Nobody mentioned lying. But the court will consider both sides of the story, and I simply do not see any strong proof that the childminder has breached the contract.
It really does need someone who is legally qualified to inspect the contract, together with any related policies, terms and conditions, and any correspondence you have saved. You may have some legal cover within your domestic insurance policy, or see a solicitor for a free 15 minute consultation to judge whether you have a reasonable case.
One thing that does seem clear is the four weeks' notice period. That remains payable whether or not you chose to use her services during that time. If the college won’t pay it, you are most likely liable as signatory to the contract.
You might be able to avoid that if you can prove that she made the arrangement unworkable, most likely by the issue over her holidays. Your problem is proving that, if for example she says you were given sufficient notice, as it then becomes your word against hers.
If the contract is as vague as you suggest, a lot will depend on legal interpretation and wrangling. Be aware this will not cost the CM a penny, as she will almost certainly have full cover and the support of a legal/debt recovery team through her childcare insurance policy.
Then again, even if they feel she has a good case, there’s some chance her insurer may just pay her themselves if they see little chance of actually recovering the money from you within a reasonable time.
Unfortunately if they do take you to court, it could end up in a court order which would damage your credit rating for a while.