And how exactly did she phrase the 'you are responsible for mending/replacing it' comment? Sounds very rude if it was put like that.
I know that au pair relationships are pretty informal as au pairs are living in your house and are not formal employees, but you still expect a bit of respect and for something like this to be addressed carefully .......it sounds like she spoke like a stroppy teenager - which I guess she might be as au pairs are often pretty young.
Q is, do you have concerns about her as an au pair? Her ability to look after the kids, to behave in an honest and trustworthy way?
Sometimes au pair relationships break down. When that happens the host sometimes uses or finds a specific incident to use as an excuse to end the contract. Au pairs are on MN all the time about this kind of thing, as are Mums with grievances, which when pressed, reveal the incident is just the tip of the iceberg.
You need to think about your au pair broadly - yes,nothing about this one off incident, but the bigger picture too. If otherwise she has been good and honest (comments about her getting your DS into trouble are worrying and odd TBH - what have you done in the past in response to these?) then it maybe possible to move forward from this one off incident. If there are other issue, maybe not.
Regardless, handle this correctly. Don't get angry. Sit her down and talk in a non confrontational way about what happened. Listen to her and try to be the mature adult in this. If you need to speak with your DC again before doing so, do so. Be aware that what they say might not be exactly what happen ended either. Think about what you want the conversation to achieve.
And if there are wider issues, be prepared to talk calmly and honestly about those too.
If you haven't previously raised any concerns, to suddenly voice loads and sack her sounds unreasonable - unless there is gros misconduct you wouldn't get away with that as any other employer......so if you need to establish clearer ground rules about her work or behaviour then do so. If you have already had a number of conversations about this kind of stuff which have failed to have an effect and this is the final straw, then be clear, factual and honest and check up on any obligations about notice, pay etc. Do the right thing.
Au pairs can do daft things (they are young adults) and an employer need some of have realistic expectations of them and be willing to help and coax them a little, as you might your own young adult child. Some can do dangerous or awful things. The distinction between these 2 different types of actions needs to be drawn by the employer who needs to be the responsible adult in the situation and make sure they do the right thing and don't over react or fail to communicate over the little things, allowing resentment to fester and build which has not been voiced.