This is exactly the kind of circumstance that you need to have in your contract - not necessarily about snow, but what happens on a day when she can't - for any reason - get to you.
And you did say in your OP that you gave her the days off, which is why some of us are saying that it appears to be a bit mean to then withhold her pay.
From my years of employing a nanny, my advice to you would be to be as generous as you possibly can. On the first two occasions I would have paid her. Eight miles in snow/ice is a long way, so I'm not surprised she was reluctant to come in. But all was OK because you were given the time off by your employer. (Were you paid?)
On the third occasion you were right to try and pre-empt the situation, but you should have had other plans in place. An alternative childminder is great, but obviously will cost you more. Unfortunately, this is one of the problems about being full time working parents! Could you not take time off work at all? I thought there was more flexibility these days from employers in terms of time needed to look after children when there are problems?
But you will really, really piss her off if you start to talk about withholding part of her bonus. As Lulu says, this is money that was supposed to be rewarding good performance and your thanks - not part of her salary.
And I still don't understand why if it's a Christmas bonus, you haven't paid it yet!